"for patterns of lengths: ", [[4, 0], [4, 0]] There all together, 76, different equivalence classes For the equivalence class of patterns, {{[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 0]]}} the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2, 3], {}, {1}, {}], [ [3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [[4, 1, 2, 3], {}, {2}, {}], [[3, 1, 2, 4], {[0, 0, 0, 1, 0]}, {2}, {}], [[1, 3, 2, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0]}, {}, {}], [[1, 4, 3, 2], {}, {2}, {}], [[3, 2, 1], {}, {1}, {}], [[1, 4, 2, 3], {}, {1}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0]}, {2}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}], [[1], {}, {}, {}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [ [2, 3, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [[2, 4, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}, {}], [[4, 1, 3, 2], {}, {1}, {}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0]}, {1}, {}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 340 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 340, 1340, 5254, 20518, 79932, 311028, 1209916, 4707964, 18330728, 71429176] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[1, 2], {[0, 0, 2]}, {2}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [2, 0], [4, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [2, 0], [4, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {4}, {}], [[3, 4, 1, 2], {}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1]}, {1, 2}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {3}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 396 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074, 176351644, 959658200] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {[[4, 0], [1, 0], [3, 0], [2, 0]], %1}, {[[4, 0], [2, 0], [1, 0], [3, 0]], %1}} %1 := [[4, 0], [3, 0], [2, 0], [1, 0]] the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[1, 3, 2], {[0, 0, 1, 1]}, {2}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0]}, {4}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1]}, {1, 2}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 1]}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 89, 380 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863, 84324840, 409471090] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 2, 0]}, {2}, {}], [[2, 3, 1], {}, {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 2, 0]}, {1, 2}, {}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {1, 2}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 0, 0, 2, 0]}, {1, 2}, {}], [[1, 2, 3], {[0, 0, 0, 1], [0, 0, 2, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 342 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[3, 2, 1], {}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 2]}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 2]}, {2}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {3}, {}], [[2, 3, 1], {[0, 0, 0, 2]}, {2}, {}], [[2, 1, 3], {[0, 0, 1, 0], [0, 0, 0, 2]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 342 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {[[4, 0], [1, 0], [3, 0], [2, 0]], %1}, {[[4, 0], [2, 0], [1, 0], [3, 0]], %1}} %1 := [[4, 0], [3, 0], [2, 0], [1, 0]] the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[1, 3, 2], {[0, 0, 1, 1]}, {2}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0]}, {4}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1]}, {1, 2}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 1]}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 89, 380 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863, 84324840, 409471090] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[3, 1, 2, 5, 4], %1, {1}, {}], [[2, 3, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2, 5, 4], %1, {1}, {}], [[2, 4, 1, 5, 3], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 4, 3], {}, {1}, {}], [[2, 4, 1, 3], {}, {1}, {}], [[1, 3, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 4, 3, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [ [4, 1, 2, 5, 3], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[2, 3, 5, 1, 4], %1, {1}, {}], [[1], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [ [1, 4, 2, 5, 3], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[2, 3, 1], {}, {}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}, {}], [[1, 4, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {}, {}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1]}, {}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1]}, {}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3], {}, {}, {}], [[2, 1, 4, 5, 3], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[4, 2, 3, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 4, 5, 2, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {4}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1]}, {}, {}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 1, 3, 5, 4], %1, {1}, {}], [[4, 1, 2, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {2}, {}], [[3, 2, 4, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 4, 5, 1, 2], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 3, 4, 1], {[0, 0, 0, 0, 1]}, {}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 2, 4, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}, {}], [[3, 1, 4, 5, 2], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 4, 1, 2], {}, {1}, {}], [[3, 1, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[4, 1, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[2, 3, 1, 5, 4], %1, {1}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {}, {}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1]}, {2}, {}], [[2, 3, 4, 1, 5], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 4, 2, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}, {}], [[2, 4, 5, 1, 3], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 4, 1, 5, 2], {[0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 1, 4, 2], {}, {1}, {}]} %1 := {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 321 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 321, 1085, 3266, 8797, 21478, 48206, 100728, 198046, 369617, 659505] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [2, 0], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [2, 0], [4, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1]}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1]}, {1, 2}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1, 2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {}, {1, 2}, {}], [[2, 4, 1, 3], {}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}], [[1, 3, 2], {}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1, 2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0]}, {2}, {}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0]}, {1, 2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 89, 382 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 89, 382, 1711, 7922, 37663, 182936, 904302, 4535994, 23034564, 118209806, 612165222] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[3, 2, 1], {}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 2]}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 2]}, {2}, {}], [[2, 3, 1], {[0, 0, 0, 2]}, {2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 87, 354 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 87, 354, 1459, 6056, 25252, 105632, 442916, 1860498, 7826120, 32956964, 138911074] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {}, {2}, {}], [[3, 1, 2], {}, {1}, {}], [[1], {}, {}, {}], [[1, 3, 2], {}, {2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, has a scheme of depth , 4 here it is: {[[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 3, 1], {}, {2}, {}], [[1], {}, {}, {}], [[4, 2, 3, 1], {}, {1}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {}, {3}, {}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0]}, {2}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {1}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}, {}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 367 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230, 13388094, 62688448, 295504025] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 1, 5, 2, 4], %3, {3}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 5, 4, 2], { [0, 0, 0, 2, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 5, 3, 1, 4], %1, {2}, {}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0]}, {4}, {}], [[3, 5, 4, 1, 2], %3, {2}, {}], [[3, 4, 1, 2], %2, {1}, {}], [[3, 1, 4, 2, 5], %3, {3}, {}], [[3, 1, 2, 4], %2, {3}, {}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2, 4], %2, {2}, {}], [[4, 1, 3, 2], %2, {3}, {}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0]}, {}, {}], [[2, 1, 4, 3, 5], %1, {3}, {}], [ [1, 4, 3, 2], {[0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {3}, {}] , [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 2, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {}, {}], [[3, 5, 4, 2, 1], {[0, 0, 0, 0, 2, 0]}, {4}, {}], [[1, 4, 2, 3], %2, {2}, {}], [[2, 1, 5, 3, 4], %1, {3}, {}], [[2, 4, 3, 1, 5], %1, {2}, {}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 4, 3, 1], {[0, 0, 0, 2, 0]}, {}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[1, 3, 2], {[0, 0, 2, 0]}, {}, {}], [[2, 5, 4, 1, 3], %1, {2}, {}], [[3, 2, 1], {}, {2}, {}]} %1 := {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]} %2 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} %3 := {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 365 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 365, 1540, 6568, 28269, 122752, 537708, 2375500, 10579400, 47469377, 214454528] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 2, 5, 4, 1], {}, {1}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[2, 4, 3, 1, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {2}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[5, 3, 4, 2, 1], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 5, 4, 3], {}, {4}, {}], [[5, 2, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [ [2, 5, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[5, 3, 4, 1, 2], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {2, 3}, {}], [[1, 4, 3, 2], {}, {3}, {}], [ [3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0]}, {}, {}], [[4, 2, 3, 1, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 5, 4, 2, 1], {}, {4}, {}], [[3, 5, 4, 1, 2], {[0, 1, 0, 0, 0, 0]}, {3}, {}], [[5, 2, 4, 1, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 5, 4, 1, 3], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}, {}], [[2, 4, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {}, {}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 363 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 363, 1507, 6241, 25721, 105485, 430767, 1752945, 7113095, 28797292, 116368938] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[2, 5, 1, 4, 3], {[0, 0, 0, 0, 0, 1]}, {2}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {2}, {}], [[2, 4, 1, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {5}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {2}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [ [3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 0, 1]}, {3}, {}], [[3, 5, 2, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[2, 5, 1, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[3, 5, 1, 4, 2], {[0, 0, 0, 0, 0, 1]}, {2}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}, {}], [[2, 4, 1, 3], {}, {}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 0, 0, 1]}, {3}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {5}, {}], [[3, 4, 1, 2], {}, {2}, {}], [[3, 1, 4, 2], {}, {3}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1]}, {2}, {}], [[3, 1, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}, {}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 338 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 338, 1318, 5110, 19770, 76466, 295810, 1144530, 4428622, 17136186, 66306722] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 4, 1, 3], {}, {1}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 4, 1, 2], {}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 1, 4, 2, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[3, 1, 2, 4], %1, {3}, {}], [[1, 3, 2, 4], %1, {2}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[3, 1, 4, 2], {}, {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0]}, {2}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0]}, {3}, {}], [[4, 2, 3, 1], %1, {1}, {}], [ [4, 2, 5, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 0, 0]}, {4}, {}], [[3, 4, 2, 1], %1, {3}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}, {}], [[3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 0, 1, 0]}, {3}, {}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 1, 0]}, {3}, {}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 337 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 337, 1299, 4910, 18228, 66640, 240550, 859295, 3043525, 10705182, 37441618] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 1, 4, 3], {}, {1}, {}], [[2, 4, 1, 3], {}, {1}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0]}, {1}, {}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}, {}], [[3, 4, 1, 2], {}, {1}, {}], [[3, 1, 4, 2], {}, {1}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[3, 1, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {1}, {}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0]}, {2}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 330 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 330, 1206, 4174, 13726, 43134, 130302, 380414, 1078270, 2978814, 8046590] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {[[4, 0], [1, 0], [3, 0], [2, 0]], %1}, {[[4, 0], [2, 0], [1, 0], [3, 0]], %1}} %1 := [[4, 0], [2, 0], [3, 0], [1, 0]] the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[0, 1, 1]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {[[4, 0], [1, 0], [3, 0], [2, 0]], %1}, {[[4, 0], [2, 0], [1, 0], [3, 0]], %1}} %1 := [[4, 0], [2, 0], [3, 0], [1, 0]] the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[0, 1, 1]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[0, 2, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 0]]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[4, 1, 3, 2], {}, {1}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {} ], [ [2, 4, 3, 1], {[0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]}, {2}, {}] , [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}, {}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {2}, {}], [[4, 2, 3, 1], {[0, 0, 1, 1, 0], [0, 0, 1, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 0, 1, 1]}, {2}, {}], [[1, 4, 3, 2], {[0, 0, 0, 1, 1]}, {2}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 367 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230, 13388094, 62688448, 295504025] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {%1, [[4, 0], [1, 0], [3, 0], [2, 0]]}, {%1, [[4, 0], [2, 0], [1, 0], [3, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {}, {}], [[1, 4, 3, 2], {}, {2}, {}], [[3, 2, 1], {}, {1}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 3, 1], {}, {2}, {}], [[1], {}, {}, {}], [[4, 1, 3, 2], {}, {1}, {}], [[4, 2, 3, 1], {}, {1}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0]}, {3}, {}], [[2, 4, 3, 1], {}, {2}, {}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0]}, {1}, {}], [ [1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [ [2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}] , [[3, 2, 4, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 368 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496, 66876096, 317809216] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 3], {}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 3, 1], {[0, 0, 2, 0]}, {1}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}, {}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 366 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368] For the equivalence class of patterns, {{[[1, 0], [3, 0], [4, 0], [2, 0]], %1}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 0], [1, 0], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], %1}, {%1, [[3, 0], [2, 0], [4, 0], [1, 0]]}, {%1, [[4, 0], [1, 0], [3, 0], [2, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 0], [1, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, has a scheme of depth , 5 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {2}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [ [3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [ [3, 5, 2, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[1, 3, 2], {[0, 1, 0, 1]}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {4}, {}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}, {}], [ [3, 1, 5, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0]}, {2}, {}], [[2, 1, 4, 3, 5], %1, {1}, {}], [[2, 4, 1, 3, 5], %1, {1}, {}], [[3, 1, 4, 2, 5], %1, {1}, {}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}, {}], [ [3, 5, 1, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}, {}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 1]}, {3}, {}], [[2, 4, 1, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}, {}], [[3, 1, 2], {[0, 1, 0, 1]}, {}, {}], [[2, 5, 1, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [ [3, 5, 1, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [[3, 1, 4, 2], {[0, 1, 0, 1, 0], [0, 1, 0, 0, 1]}, {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[0, 1, 0, 1, 0], [0, 1, 0, 0, 1]}, {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {2}, {}], [ [2, 5, 1, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 1]}, {2}, {}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [ [4, 1, 5, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[4, 5, 2, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 4, 1, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [ [4, 5, 1, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}]} %1 := {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 336 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 336, 1290, 4870, 18164, 67234, 247786, 911120, 3346618, 12286942, 45104548] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 2, 5, 4, 1], {}, {1}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 5, 4, 3], {}, {4}, {}], [[1, 4, 3, 2], {}, {3}, {}], [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}, {}], [[2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[1, 3, 2, 4], %1, {1}, {}], [[1, 4, 2, 3], %1, {3}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], %1, {1}, {}], [[2, 4, 1, 3], %1, {3}, {}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}, {}], [ [2, 1, 5, 3, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {}, {3}, {}]} %1 := {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 366 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}, {}], [[3, 4, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [ [2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [ [3, 1, 4, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 5, 4, 2], {[0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [ [3, 1, 5, 2, 4], {[0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {[0, 0, 1, 1, 0]}, {3}, {}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0]}, {3}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 1, 1, 0]}, {4}, {}], [[3, 2, 5, 4, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0]}, {1}, {}], [[4, 2, 5, 3, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 87, 352 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 87, 352, 1434, 5861, 24019, 98677, 406291, 1676009, 6924618, 28646875, 118638038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 5, 3, 1], %1, {1}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0]}, {2}, {}], [[2, 1, 4, 3, 5], %2, {1}, {}], [[2, 4, 1, 3, 5], %2, {1}, {}], [[3, 1, 4, 2, 5], %2, {1}, {}], [ [4, 1, 5, 2, 3], {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {1}, {}], [ [3, 1, 5, 4, 2], {[0, 1, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}, {}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}, {}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [ [2, 5, 1, 4, 3], {[0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}, {}], [ [2, 5, 1, 3, 4], {[0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}, {}], [[3, 5, 2, 4, 1], %1, {1}, {}], [[3, 5, 1, 4, 2], {[0, 1, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 3, 2], {[0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [ [4, 5, 1, 3, 2], {[0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[4, 5, 2, 3, 1], %1, {1}, {}], [[2, 4, 1, 3], {[0, 1, 0, 1, 0], [0, 0, 1, 1, 0]}, {}, {}], [[1, 3, 2], {[0, 1, 1, 0]}, {}, {}], [[3, 4, 1, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0]}, {1}, {}], [ [3, 5, 1, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}, {}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}, {}], [[3, 1, 4, 2], {[0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 0]}, {4}, {}], [[2, 1, 4, 3], {[0, 1, 0, 1, 0], [0, 0, 1, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 1, 1, 0]}, {}, {}], [[3, 4, 1, 2], {[0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {4}, {}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {2}, {}]} %1 := {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} %2 := {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 338 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 338, 1318, 5106, 19718, 76066, 293398, 1131794, 4366374, 16846018, 64995254] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, has a scheme of depth , 5 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {2}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}, {}], [ [3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [ [3, 5, 2, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[1, 3, 2], {[0, 1, 0, 1]}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {4}, {}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}, {}], [ [3, 1, 5, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0]}, {2}, {}], [[2, 1, 4, 3, 5], %1, {1}, {}], [[2, 4, 1, 3, 5], %1, {1}, {}], [[3, 1, 4, 2, 5], %1, {1}, {}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}, {}], [ [3, 5, 1, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}, {}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 1]}, {3}, {}], [[2, 4, 1, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}, {}], [[3, 1, 2], {[0, 1, 0, 1]}, {}, {}], [[2, 5, 1, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [ [3, 5, 1, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}, {}], [[3, 1, 4, 2], {[0, 1, 0, 1, 0], [0, 1, 0, 0, 1]}, {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[0, 1, 0, 1, 0], [0, 1, 0, 0, 1]}, {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {2}, {}], [ [2, 5, 1, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 1]}, {2}, {}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [ [4, 1, 5, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[4, 5, 2, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}], [[3, 4, 1, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [ [4, 5, 1, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}, {}]} %1 := {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 336 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 336, 1290, 4870, 18164, 67234, 247786, 911120, 3346618, 12286942, 45104548] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 2, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 2, 0, 0]}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 3, 1], {[0, 0, 2, 0]}, {1}, {}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 342 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [4, 0], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {}, {1, 2}, {}], [[2, 4, 1, 3], {}, {1, 2}, {}], [[2, 3, 1], {}, {}, {}], [[1, 3, 2], {}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 1, 5, 2, 4], %3, {3}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 5, 4, 2], { [0, 0, 0, 2, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 5, 3, 1, 4], %1, {2}, {}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0]}, {4}, {}], [[3, 5, 4, 1, 2], %3, {2}, {}], [[3, 4, 1, 2], %2, {1}, {}], [[3, 1, 4, 2, 5], %3, {3}, {}], [[3, 1, 2, 4], %2, {3}, {}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2, 4], %2, {2}, {}], [[4, 1, 3, 2], %2, {3}, {}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0]}, {}, {}], [[2, 1, 4, 3, 5], %1, {3}, {}], [ [1, 4, 3, 2], {[0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {3}, {}] , [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 2, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {}, {}], [[3, 5, 4, 2, 1], {[0, 0, 0, 0, 2, 0]}, {4}, {}], [[1, 4, 2, 3], %2, {2}, {}], [[2, 1, 5, 3, 4], %1, {3}, {}], [[2, 4, 3, 1, 5], %1, {2}, {}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 4, 3, 1], {[0, 0, 0, 2, 0]}, {}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[1, 3, 2], {[0, 0, 2, 0]}, {}, {}], [[2, 5, 4, 1, 3], %1, {2}, {}], [[3, 2, 1], {}, {2}, {}]} %1 := {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]} %2 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} %3 := {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 365 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 365, 1540, 6568, 28269, 122752, 537708, 2375500, 10579400, 47469377, 214454528] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 4, 1, 3], {}, {1}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 4, 1, 2], {}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 1, 4, 2, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[3, 1, 2, 4], %1, {3}, {}], [[1, 3, 2, 4], %1, {2}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[3, 1, 4, 2], {}, {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0]}, {2}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0]}, {3}, {}], [[4, 2, 3, 1], %1, {1}, {}], [ [4, 2, 5, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 0, 0]}, {4}, {}], [[3, 4, 2, 1], %1, {3}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}, {}], [[3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 0, 1, 0]}, {3}, {}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 1, 0]}, {3}, {}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 86, 337 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 86, 337, 1299, 4910, 18228, 66640, 240550, 859295, 3043525, 10705182, 37441618] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 2, 5, 4, 1], {}, {1}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}, {}], [[2, 4, 3, 1, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {2}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[5, 3, 4, 2, 1], {[0, 0, 0, 1, 0, 0]}, {4}, {}], [[2, 1, 5, 4, 3], {}, {4}, {}], [[5, 2, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [ [2, 5, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}, {}], [[5, 3, 4, 1, 2], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {2, 3}, {}], [[1, 4, 3, 2], {}, {3}, {}], [ [3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0]}, {}, {}], [[4, 2, 3, 1, 5], {[0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 5, 4, 2, 1], {}, {4}, {}], [[3, 5, 4, 1, 2], {[0, 1, 0, 0, 0, 0]}, {3}, {}], [[5, 2, 4, 1, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 5, 4, 1, 3], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}, {}], [[2, 4, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {}, {}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 363 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 363, 1507, 6241, 25721, 105485, 430767, 1752945, 7113095, 28797292, 116368938] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 0], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], %1}, {[[3, 0], [2, 0], [4, 0], [1, 0]], %1}, {%1, [[4, 0], [1, 0], [3, 0], [2, 0]]}, {%1, [[4, 0], [2, 0], [1, 0], [3, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 0]]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {}, {}], [[1, 4, 3, 2], {}, {2}, {}], [[3, 2, 1], {}, {1}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 3, 1], {}, {2}, {}], [[1], {}, {}, {}], [[4, 1, 3, 2], {}, {1}, {}], [[4, 2, 3, 1], {}, {1}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0]}, {3}, {}], [[2, 4, 3, 1], {}, {2}, {}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0]}, {1}, {}], [ [1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}] , [ [2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}] , [[3, 2, 4, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 368 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496, 66876096, 317809216] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 3], {}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, {{[[1, 0], [4, 0], [2, 0], [3, 0]], %1}, {[[2, 0], [3, 0], [1, 0], [4, 0]], %1}, {%1, [[2, 0], [4, 0], [3, 0], [1, 0]]}, {%1, [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [1, 0], [4, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 0], [1, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[3, 2, 5, 4, 1], {}, {1}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 5, 4, 3], {}, {4}, {}], [[1, 4, 3, 2], {}, {3}, {}], [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}, {}], [[2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[1, 3, 2, 4], %1, {1}, {}], [[1, 4, 2, 3], %1, {3}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], %1, {1}, {}], [[2, 4, 1, 3], %1, {3}, {}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}, {}], [ [2, 1, 5, 3, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {}, {3}, {}]} %1 := {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 88, 366 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[0, 2, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 90, 394 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [2, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [3, 0], [2, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 0], [3, 0]]}, has a scheme of depth , 5 here it is: {[[1, 3, 2], {}, {}, {}], [[1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {3}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {}, {}, {}], [[2, 1, 3], {}, {}, {}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}, {}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}, {}], [[3, 4, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 4, 1], {}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[2, 1, 4, 3], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [ [2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [ [3, 1, 4, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 5, 4, 2], {[0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0]}, {4}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[4, 1, 5, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}, {}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [ [3, 1, 5, 2, 4], {[0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}, {}], [[2, 4, 3, 1], {[0, 0, 1, 1, 0]}, {3}, {}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0]}, {3}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 1, 1, 0]}, {4}, {}], [[3, 2, 5, 4, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0]}, {1}, {}], [[4, 2, 5, 3, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 6, 22, 87, 352 Using the scheme, the first, , 16, terms are [1, 1, 2, 6, 22, 87, 352, 1434, 5861, 24019, 98677, 406291, 1676009, 6924618, 28646875, 118638038] Out of a total of , 76, cases 43, were successful and , 33, failed Success Rate: , 0.566 Here are the failures {{{%22, %2}, {%19, %4}, {%19, %2}, {%4, %21}, {%4, %20}, {%2, %21}, {%2, %20}}, {{%24, %23}}, {{%18, %17}}, {{%4, %2}}, {{%24, %5}, {%24, %1}, {%6, %23}, {%3, %23}}, {{%24, %4}, {%24, %2}, {%4, %23}, {%2, %23}}, {{%24, %14}, {%24, %12}, {%24, %10}, {%16, %23}, {%15, %23}, {%9, %23}, {%8, %23}}, {{%24, %13}, {%24, %12}, {%24, %10}, {%16, %23}, {%15, %23}, {%9, %23}, {%8, %23}}, {{%24, %7}, {%11, %23}}, {{%24, %17}, {%18, %23}}, { {%22, %4}, {%22, %2}, {%19, %4}, {%4, %21}, {%4, %20}, {%2, %21}, {%2, %20} }, {{%22, %7}, {%19, %7}, {%11, %21}, {%11, %20}}, {{%22, %17}, {%18, %21}, {%18, %20}, {%19, %17}}, {{%18, %6}, {%18, %3}, {%5, %17}, {%1, %17}}, {{%18, %11}, {%7, %17}}, {{%18, %5}, {%18, %1}, {%6, %17}, {%3, %17}}, {{%18, %4}, {%18, %2}, {%4, %17}, {%2, %17}}, {{%18, %14}, {%18, %12}, {%18, %10}, {%16, %17}, {%15, %17}, {%9, %17}, {%8, %17}}, {{%18, %13}, {%18, %12}, {%18, %10}, {%16, %17}, {%15, %17}, {%9, %17}, {%8, %17}}, {{%18, %7}, {%11, %17}}, {{%16, %15}, {%9, %8}, {%14, %13}, {%12, %10}}, {{%16, %9}, {%15, %8}, {%14, %12}, {%13, %10}}, { {%16, %4}, {%15, %2}, {%4, %8}, {%4, %13}, {%4, %12}, {%14, %2}, {%2, %10}} , {{%16, %8}, {%15, %9}, {%14, %10}, {%13, %12}}, {{%16, %7}, {%15, %7}, {%11, %14}, {%11, %13}, {%11, %12}, {%11, %10}, {%8, %7}}, {{%6, %11}, {%11, %3}, {%5, %7}, {%7, %1}}, {{%6, %4}, {%6, %2}, {%5, %4}, {%5, %2}, {%4, %3}, {%4, %1}, {%2, %1}}, {{%6, %3}, {%5, %1}}, {{%6, %7}, {%11, %5}, {%11, %1}, {%3, %7}}, {{%11, %4}, {%11, %2}, {%4, %7}, {%2, %7}}, {{%15, %2}, {%9, %2}, {%4, %8}, {%4, %13}, {%4, %12}, {%14, %2}, {%2, %10}} , {{%15, %7}, {%11, %14}, {%11, %13}, {%11, %12}, {%11, %10}, {%9, %7}, {%8, %7}}, {{%6, %2}, {%5, %4}, {%5, %2}, {%4, %3}, {%4, %1}, {%2, %3}, {%2, %1}}} %1 := [[4, 0], [1, 0], [2, 0], [3, 0]] %2 := [[3, 0], [1, 0], [4, 0], [2, 0]] %3 := [[3, 0], [2, 0], [1, 0], [4, 0]] %4 := [[2, 0], [4, 0], [1, 0], [3, 0]] %5 := [[2, 0], [3, 0], [4, 0], [1, 0]] %6 := [[1, 0], [4, 0], [3, 0], [2, 0]] %7 := [[3, 0], [4, 0], [1, 0], [2, 0]] %8 := [[3, 0], [1, 0], [2, 0], [4, 0]] %9 := [[2, 0], [3, 0], [1, 0], [4, 0]] %10 := [[4, 0], [2, 0], [1, 0], [3, 0]] %11 := [[2, 0], [1, 0], [4, 0], [3, 0]] %12 := [[4, 0], [1, 0], [3, 0], [2, 0]] %13 := [[3, 0], [2, 0], [4, 0], [1, 0]] %14 := [[2, 0], [4, 0], [3, 0], [1, 0]] %15 := [[1, 0], [4, 0], [2, 0], [3, 0]] %16 := [[1, 0], [3, 0], [4, 0], [2, 0]] %17 := [[4, 0], [2, 0], [3, 0], [1, 0]] %18 := [[1, 0], [3, 0], [2, 0], [4, 0]] %19 := [[2, 0], [1, 0], [3, 0], [4, 0]] %20 := [[4, 0], [3, 0], [1, 0], [2, 0]] %21 := [[3, 0], [4, 0], [2, 0], [1, 0]] %22 := [[1, 0], [2, 0], [4, 0], [3, 0]] %23 := [[4, 0], [3, 0], [2, 0], [1, 0]] %24 := [[1, 0], [2, 0], [3, 0], [4, 0]] {{{%22, %2}, {%19, %4}, {%19, %2}, {%4, %21}, {%4, %20}, {%2, %21}, {%2, %20}}, {{%24, %23}}, {{%18, %17}}, {{%4, %2}}, {{%24, %5}, {%24, %1}, {%6, %23}, {%3, %23}}, {{%24, %4}, {%24, %2}, {%4, %23}, {%2, %23}}, {{%24, %14}, {%24, %12}, {%24, %10}, {%16, %23}, {%15, %23}, {%9, %23}, {%8, %23}}, {{%24, %13}, {%24, %12}, {%24, %10}, {%16, %23}, {%15, %23}, {%9, %23}, {%8, %23}}, {{%24, %7}, {%11, %23}}, {{%24, %17}, {%18, %23}}, { {%22, %4}, {%22, %2}, {%19, %4}, {%4, %21}, {%4, %20}, {%2, %21}, {%2, %20} }, {{%22, %7}, {%19, %7}, {%11, %21}, {%11, %20}}, {{%22, %17}, {%18, %21}, {%18, %20}, {%19, %17}}, {{%18, %6}, {%18, %3}, {%5, %17}, {%1, %17}}, {{%18, %11}, {%7, %17}}, {{%18, %5}, {%18, %1}, {%6, %17}, {%3, %17}}, {{%18, %4}, {%18, %2}, {%4, %17}, {%2, %17}}, {{%18, %14}, {%18, %12}, {%18, %10}, {%16, %17}, {%15, %17}, {%9, %17}, {%8, %17}}, {{%18, %13}, {%18, %12}, {%18, %10}, {%16, %17}, {%15, %17}, {%9, %17}, {%8, %17}}, {{%18, %7}, {%11, %17}}, {{%16, %15}, {%9, %8}, {%14, %13}, {%12, %10}}, {{%16, %9}, {%15, %8}, {%14, %12}, {%13, %10}}, { {%16, %4}, {%15, %2}, {%4, %8}, {%4, %13}, {%4, %12}, {%14, %2}, {%2, %10}} , {{%16, %8}, {%15, %9}, {%14, %10}, {%13, %12}}, {{%16, %7}, {%15, %7}, {%11, %14}, {%11, %13}, {%11, %12}, {%11, %10}, {%8, %7}}, {{%6, %11}, {%11, %3}, {%5, %7}, {%7, %1}}, {{%6, %4}, {%6, %2}, {%5, %4}, {%5, %2}, {%4, %3}, {%4, %1}, {%2, %1}}, {{%6, %3}, {%5, %1}}, {{%6, %7}, {%11, %5}, {%11, %1}, {%3, %7}}, {{%11, %4}, {%11, %2}, {%4, %7}, {%2, %7}}, {{%15, %2}, {%9, %2}, {%4, %8}, {%4, %13}, {%4, %12}, {%14, %2}, {%2, %10}} , {{%15, %7}, {%11, %14}, {%11, %13}, {%11, %12}, {%11, %10}, {%9, %7}, {%8, %7}}, {{%6, %2}, {%5, %4}, {%5, %2}, {%4, %3}, {%4, %1}, {%2, %3}, {%2, %1}}} %1 := [[4, 0], [1, 0], [2, 0], [3, 0]] %2 := [[3, 0], [1, 0], [4, 0], [2, 0]] %3 := [[3, 0], [2, 0], [1, 0], [4, 0]] %4 := [[2, 0], [4, 0], [1, 0], [3, 0]] %5 := [[2, 0], [3, 0], [4, 0], [1, 0]] %6 := [[1, 0], [4, 0], [3, 0], [2, 0]] %7 := [[3, 0], [4, 0], [1, 0], [2, 0]] %8 := [[3, 0], [1, 0], [2, 0], [4, 0]] %9 := [[2, 0], [3, 0], [1, 0], [4, 0]] %10 := [[4, 0], [2, 0], [1, 0], [3, 0]] %11 := [[2, 0], [1, 0], [4, 0], [3, 0]] %12 := [[4, 0], [1, 0], [3, 0], [2, 0]] %13 := [[3, 0], [2, 0], [4, 0], [1, 0]] %14 := [[2, 0], [4, 0], [3, 0], [1, 0]] %15 := [[1, 0], [4, 0], [2, 0], [3, 0]] %16 := [[1, 0], [3, 0], [4, 0], [2, 0]] %17 := [[4, 0], [2, 0], [3, 0], [1, 0]] %18 := [[1, 0], [3, 0], [2, 0], [4, 0]] %19 := [[2, 0], [1, 0], [3, 0], [4, 0]] %20 := [[4, 0], [3, 0], [1, 0], [2, 0]] %21 := [[3, 0], [4, 0], [2, 0], [1, 0]] %22 := [[1, 0], [2, 0], [4, 0], [3, 0]] %23 := [[4, 0], [3, 0], [2, 0], [1, 0]] %24 := [[1, 0], [2, 0], [3, 0], [4, 0]] "for patterns of lengths: ", [[4, 1], [4, 0]] There all together, 576, different equivalence classes For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 4 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0]}, {3}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, {{%1, [[4, 0], [1, 0], [2, 1], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {%1, [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {%1, [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 1], [3, 0], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 1], [2, 0], [3, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[4, 0], [3, 0], [2, 1], [1, 0]]}, {%1, [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {%1, [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {%1, [[1, 0], [2, 1], [3, 0], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}] , [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, {{%1, [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {%1, [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 1], [1, 0], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {%1, [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {%1, [[4, 0], [3, 1], [2, 0], [1, 0]]}, {%1, [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {%1, [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2, 3], {}, {1}, {}], [[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {%1, [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}] , [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 30, 61 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 2, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[4, 0], [2, 0], [3, 0], [1, 0]] the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 1, 1]}, {1}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}] , [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1], [1, 0, 1, 0]}, {2}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 1]}, {1}, {}], [[1, 2], {[1, 0, 1]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[4, 0], [3, 0], [2, 0], [1, 0]] the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 31, 66 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, 3291] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0], [2, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2, 3], {[1, 1, 0, 0]}, {1}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {%1, [[1, 0], [2, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[1, 0], [2, 0], [3, 0], [4, 0]] the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 4 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0]}, {3}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {%1, [[1, 0], [2, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[1, 0], [3, 0], [2, 0], [4, 0]] the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {%1, [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [3, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, {{%1, [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {%1, [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [2, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2, 3], {}, {1}, {}], [[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1], [1, 0, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 1]}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 1]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 1, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[2, 0], [1, 0], [4, 0], [3, 0]] the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 1, 1]}, {1}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[1, 0], [2, 0], [3, 0], [4, 0]] the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0], [2, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1], [1, 0, 1, 0]}, {2}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 1]}, {1}, {}], [[1, 2], {[1, 0, 1]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[1, 0], [3, 0], [2, 0], [4, 0]] the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [3, 0], [2, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 1], [2, 0], [1, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [2, 0], [1, 1], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[1, 0], [2, 0], [3, 0], [4, 0]] the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, {%1, [[1, 0], [4, 0], [3, 0], [2, 1]]}, {%1, [[1, 0], [4, 1], [3, 0], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 1], [3, 0], [4, 0], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 0], [3, 0], [4, 1], [1, 0]]}, {%1, [[3, 1], [2, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [2, 0], [1, 1], [4, 0]]}} %1 := [[2, 0], [1, 0], [4, 0], [3, 0]] the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [1, 1], [2, 0], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[1, 0], [3, 0], [2, 0], [4, 0]] the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 1, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[2, 0], [1, 0], [4, 0], [3, 0]] the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 2, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1], [1, 0, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 1]}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 1]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 1, 0]}, {1}, {}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 72 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [1, 1], [2, 0]]}, {%1, [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}] , [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 30, 61 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 0, 1], [0, 0, 2, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [ [3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {1}, {}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {1}, {}], [ [2, 3, 1, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}, {}] , [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 25, 25 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 25, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [1, 1], [2, 0]]}, {%1, [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 0], [1, 1]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 1], [2, 0], [3, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [2, 0], [4, 1]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 1], [3, 0], [2, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[1, 0], [2, 0], [3, 0], [4, 0]] the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {%1, [[2, 0], [1, 0], [4, 0], [3, 1]]}, {%1, [[2, 0], [1, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [4, 0], [3, 0]]}, {%1, [[2, 1], [1, 0], [4, 0], [3, 0]]}} %1 := [[1, 0], [3, 0], [2, 0], [4, 0]] the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2, 3], {[1, 1, 0, 0]}, {1}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 1, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, {{%1, [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {%1, [[4, 0], [2, 1], [3, 0], [1, 0]]}, {%1, [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {%1, [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 1, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [1, 1], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 1], [4, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 0], [3, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [4, 1], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 1], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 1], [1, 0], [4, 0], [3, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}] , [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 30, 61 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 1], [1, 0], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 1], [4, 0], [3, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [3, 0], [4, 0], [1, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [2, 0], [1, 0], [4, 1]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [3, 0], [4, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[2, 1], {}, {1}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[0, 0, 1]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1], [1, 0, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 1]}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 1]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 2, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [4, 0], [1, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 1], [2, 0], [1, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 1], [1, 0], [3, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 1], [3, 0], [4, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 1], [4, 0], [2, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [1, 0], [4, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 1]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [1, 0], [2, 0], [4, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[4, 0], [2, 0], [3, 1], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [2, 1], [3, 0], [1, 0]]}, {%1, [[4, 0], [2, 1], [3, 0], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [2, 1], [4, 0]]}, {%1, [[1, 0], [3, 0], [2, 1], [4, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [3, 1], [2, 0], [4, 0]]}, {%1, [[1, 0], [3, 1], [2, 0], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [3, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 1, 1]}, {1}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1], [1, 0, 1, 0]}, {2}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 1]}, {1}, {}], [[1, 2], {[1, 0, 1]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}], [[2, 1], {[0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[0, 0, 1, 1]}, {1}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2, 3], {}, {1}, {}], [[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[1, 0], [4, 0], [3, 1], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [3, 1], [4, 0], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [2, 1], [1, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [1, 0], [2, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 31, 66 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, 3291] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 1, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 1], [3, 0], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 1], [2, 0], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 0], [4, 1]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 0], [1, 1]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 1, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, { {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2, 3], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}, {}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[1, 2], {[0, 2, 0]}, {}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[4, 0], [2, 0], [3, 0], [1, 0]] the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {}, {2}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 81 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 0, 0, 1], [1, 0, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 1]}, {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 1]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[4, 0], [3, 0], [2, 0], [1, 0]] the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 31, 66 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, 3291] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {%1, [[4, 0], [3, 1], [1, 0], [2, 0]]}, {%1, [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {%1, [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[2, 1], {[0, 1, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[3, 0], [4, 0], [2, 1], [1, 0]]}, {%1, [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {%1, [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {%1, [[2, 0], [1, 0], [3, 1], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 1], [2, 0], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [4, 0], [2, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [1, 1], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 1], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 1], [4, 0], [3, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [4, 1], [1, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 1]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 0], [3, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [4, 1], [2, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [4, 0], [2, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [1, 1], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 1], [1, 0], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 1], [4, 0], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [2, 0], [4, 1], [1, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [3, 0], [1, 1], [4, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[2, 3, 1], {}, {2}, {}], [[1], {}, {}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 40, 114 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 40, 114, 324, 920, 2612, 7416, 21056, 59784, 169744, 481952, 1368400] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[3, 2, 1], {}, {1}, {}], [[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}, {}] , [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[1, 0], [3, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {%1, [[1, 0], [2, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[4, 0], [2, 0], [3, 0], [1, 0]] the member , {[[1, 0], [3, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 1, 0]}, {1}, {}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 72 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[1, 0], [3, 0], [2, 0], [4, 0]] the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}] , [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 30, 61 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {%1, [[1, 0], [2, 0], [4, 1], [3, 0]]}, {%1, [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[4, 0], [3, 0], [2, 0], [1, 0]] the member , {[[1, 0], [2, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}, {}], [[1, 2, 4, 3], { [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {1}, {}], [ [2, 3, 4, 1], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}, {}] , [[1, 2, 3], {[0, 0, 0, 1], [0, 0, 2, 0]}, {}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [ [3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {1}, {}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {1}, {}], [ [2, 3, 1, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}, {}] , [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 25, 25 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 25, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[3, 0], [4, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[2, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [2, 0], [3, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [3, 0], [2, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [2, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [4, 0], [1, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[2, 0], [1, 0], [4, 0], [3, 0]] the member , {[[3, 0], [4, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 1, 0]}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {}, {2}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 4 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0]}, {3}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, has a scheme of depth , 3 here it is: {[[2, 1], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[3, 2, 1], {[0, 1, 1, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[1, 2], {[1, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, {{%1, [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {%1, [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 0], [4, 0], [2, 1], [1, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [2, 1], [4, 0], [3, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 0], [1, 0], [3, 1], [4, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [3, 1], [1, 0], [2, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[2, 0], [3, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[3, 0], [2, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {%1, [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, {{%1, [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {%1, [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[2, 0], [4, 0], [1, 0], [3, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[3, 0], [1, 0], [4, 0], [2, 0]] the member , {[[3, 0], [1, 0], [4, 0], [2, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 33, 82 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945] For the equivalence class of patterns, { {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2, 3], {}, {1}, {}], [[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[1, 0], [3, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[2, 0], [3, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [4, 0], [3, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[2, 0], [1, 0], [4, 0], [3, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[3, 0], [4, 0], [1, 0], [2, 0]] the member , {[[2, 0], [1, 0], [4, 0], [3, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 3 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}, {}], [[2, 1], {[0, 1, 0]}, {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, {{%1, [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {%1, [[4, 0], [3, 0], [1, 1], [2, 0]]}, {%1, [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[3, 0], [1, 0], [4, 0], [2, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {%1, [[2, 1], [1, 0], [3, 0], [4, 0]]}} %1 := [[2, 0], [4, 0], [1, 0], [3, 0]] the member , {[[2, 0], [4, 0], [1, 0], [3, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {[1, 1, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {%1, [[1, 0], [3, 0], [4, 1], [2, 0]]}, {%1, [[1, 0], [4, 0], [2, 0], [3, 1]]}, {%1, [[2, 1], [3, 0], [1, 0], [4, 0]]}, {%1, [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[4, 0], [3, 0], [2, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} %1 := [[1, 0], [2, 0], [3, 0], [4, 0]] the member , {[[4, 0], [3, 0], [2, 0], [1, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0], [2, 0, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[4, 0], [1, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[1, 0], [4, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[3, 0], [2, 0], [1, 0], [4, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [4, 0], [1, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 34, 89 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229] For the equivalence class of patterns, { {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [3, 0], [4, 1], [2, 0]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[1, 0], [4, 0], [2, 0], [3, 1]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[2, 1], [3, 0], [1, 0], [4, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [1, 1], [2, 0], [4, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [4, 1], [3, 0], [1, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[3, 1], [2, 0], [4, 0], [1, 0]]}} the member , {[[4, 0], [3, 0], [1, 0], [2, 0]], [[4, 0], [2, 0], [1, 1], [3, 0]]}, has a scheme of depth , 2 here it is: {[[], {}, {}, {}], [[1], {}, {}, {}], [[2, 1], {[0, 1, 0]}, {1}, {}], [[1, 2], {}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 14, 42, 132 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845] For the equivalence class of patterns, { {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[3, 0], [1, 0], [2, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [4, 0], [2, 0], [3, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[2, 0], [4, 0], [3, 0], [1, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[4, 0], [1, 0], [3, 0], [2, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[3, 0], [2, 0], [4, 0], [1, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[4, 0], [2, 0], [1, 0], [3, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[2, 0], [3, 0], [1, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, has a scheme of depth , 3 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {2}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 32, 74 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031] For the equivalence class of patterns, { {[[2, 0], [1, 0], [3, 0], [4, 0]], [[3, 0], [4, 1], [2, 0], [1, 0]]}, {[[2, 0], [1, 0], [3, 0], [4, 0]], [[4, 0], [3, 0], [1, 0], [2, 1]]}, {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[1, 0], [2, 0], [4, 0], [3, 1]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[1, 0], [2, 0], [4, 1], [3, 0]]}, {[[3, 0], [4, 0], [2, 0], [1, 0]], [[2, 0], [1, 1], [3, 0], [4, 0]]}, {[[4, 0], [3, 0], [1, 0], [2, 0]], [[2, 1], [1, 0], [3, 0], [4, 0]]}} the member , {[[1, 0], [2, 0], [4, 0], [3, 0]], [[4, 0], [3, 0], [1, 1], [2, 0]]}, has a scheme of depth , 4 here it is: {[[1, 2], {}, {}, {}], [[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}, {}], [[2, 1], {[1, 0, 0]}, {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}, {}] , [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}, {}]} Naively, we would expect the sequence to begin , 1, 1, 2, 5, 13, 30, 61 Using the scheme, the first, , 16, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290]