0213_0231_0312_1320_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 50, 131, 328, 793, 1866, 4299, 9740, 21773, 48142, 105487, 229392, 495633, 1064978, 2277395, 4849684, 10289173, 21757974, 45875223, 96469016, 202375193, 423624730, 884998171, 1845493788, 3841982493, 7985954846, 16575889439, 34359738400, 71135395873, 147102629922, 303868936227, 627065225252, 1292785156133, 2662879723558, 5480378269735, 11269994184744, 23158463660073, 47553877901354, 97581656965163, 200111116255276, 410117837160493, 840026883620910, 1719636185841711, 3518437208883248, 7195204092166193, 14707067533131826, 30047453763862579, 61361544922923060, 125256364636241973, 255579278853275702, 521291656868134967, 1062849512059437112, 2166231420765208633, 4413527634823086138, 8989184856231510075, 18302628885633695804, 37253776117608742973, 75804588927900188734, 154203251241165783103, 313594649253062377536, 637565592047586377793, 1295883771178096001090, 2633272716522038493251, 5349555781375769968708, 10865132259414925901893, 22062305912156623732806, 44788694610966791323719, 90905554795240670363720, 184467440737095516160073, 374247543767419383185482, 759120412121295468101707, 1539491473415504339664972, 3121484245176835486253133, 6327971087045324586352718, 12825947367473956400398415, 25991905121714527256182864, 52663831016962283423137873, 106687703580991024667820114, 216095490256114964978729043, 437631146700495761243635796, 886142625777523185059627093, 1794045916308109695263965270, 3631613162122346040817352791, 7350268983256945382213550168, 14874623284538397365584789593, 30097417205125807933484957786, 60891175682349642271600672859, 123175033908895337352462860380, 249135432906182780323448750173, 503841595989149771883943559262, 1018824652331867966241979236447, 2059932225370872777432142708832, 4164430292156019244760653889633, 8417992267140585869314044723298, 17014247899938266498213563334755, 34385022531190722515598074445924, 69483098525009824069538044444773
Generating function in Maple syntax:
(x^6-5*x^3+9*x^2-5*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
\frac{x^{6}-5 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 - 5*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^2*F(x)+x^6-5*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-5 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/16*(9*n-23)*2^n+n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(9 n -23\right) 2^{n}}{16}+n +1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 50
a(6) = 131
a(n+2) = -4*a(n)+4*a(n+1)-1+n, n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 131
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-1+n, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22477/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[11,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[11,x]*F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]*F[12,x]*F[17,x]
F[11,x] = x
F[12,x] = F[13,x]+F[15,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[11,x]*F[13,x]
F[15,x] = F[16,x]
F[16,x] = F[11,x]*F[12,x]*F[13,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[13,x]*F[17,x]
F[20,x] = F[21,x]
F[21,x] = F[11,x]*F[22,x]
F[22,x] = F[23,x]+F[34,x]
F[23,x] = F[24,x]+F[9,x]
F[24,x] = F[25,x]*F[33,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[12,x]
F[29,x] = F[27,x]*F[30,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[11,x]*F[30,x]
F[33,x] = F[18,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[17,x]*F[27,x]
F[36,x] = F[31,x]*F[7,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{15}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{34}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{33}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right) F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{27}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right) F_{7}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_11(x)*F_4(x))
Eq(F_4(x), F_20(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_11(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x)*F_12(x)*F_17(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x) + F_15(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_11(x)*F_13(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_11(x)*F_12(x)*F_13(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_13(x)*F_17(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_34(x))
Eq(F_23(x), F_24(x) + F_9(x))
Eq(F_24(x), F_25(x)*F_33(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_12(x))
Eq(F_29(x), F_27(x)*F_30(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_11(x)*F_30(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_17(x)*F_27(x))
Eq(F_36(x), F_31(x)*F_7(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], 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