0132_0213_1230_2031
Counting sequence:
1, 1, 2, 6, 20, 64, 197, 589, 1726, 4986, 14252, 40408, 113825, 318921, 889538, 2471462, 6843268, 18891368, 52010741, 142846325, 391462766, 1070636906, 2922793052, 7965674256, 21675667105, 58897685329, 159824990882, 433163677734, 1172617518836, 3170956767856, 8566104732677, 23118644992093, 62337840345790, 167947617761466, 452115226320908, 1216175956939432, 3269136113972993, 8781624890619033, 23574192593921186, 63245922475161350, 169580029517715556, 454438560504110936, 1217152380508665077, 3258324372011410181, 8718321379775426990, 23316835908743769194, 62332274125956885692, 166560053665334015904, 444888000677762723521, 1187844222592629904801, 3170325375962764056002, 8458433757652955278278, 22559200745195728704980, 60146541170160261115936, 160308315993738346016453, 427134713804147890752877, 1137736853169465473766718, 3029642621963329226719098, 8065250313746831612238956, 21464719446096733963646200, 57110682103975314230929121, 151914037977304741361588841, 403989791082932381225491202, 1074083701924995370353346406, 2854988055397566066560767300, 7587032319730731862007587784, 20157837729478196175292597877, 53545515490291514483742739349, 142204083780475732859302903406, 377583826346786148689034931946, 1002373291707754665853261421852, 2660496647624443152321118535088, 7060172551261598603368821762145, 18732228107600458555773099364081, 49692077175764070564870025171298, 131798492530924527127484827720422, 349511302346483538285700440102068, 926704631185715394532285521252688, 2456702339312756535641108137687301, 6511732414381646572355255749280125, 17257385238617366364154448422721086, 45728764278196909697277361197346746, 121155040191367551058566872244559052, 320946413105361851264977027034234056, 850088236957692137718302829579678785, 2251320354457180858973415427625217849, 5961474958649275195347845148892462562, 15783908861507452698225951433731706694, 41785062513688082056330600413112046116, 110604907002984982970092401179530815416, 292735732577736436194085273704995183477, 774686884654204844131893097024795476773, 2049872629074350082418444202501166484910, 5423489531713280843250700429952679522026, 14347723845809326781492829526258281843068, 37952507115801767063568724452831668688192, 100381144952113342762061900077754329809793, 265472144982003298718797307892144391071553, 702007594267774297529982983139860712361346, 1856186333401487788781866307170000302351750, 4907486188403315909412003614338936278109076
Generating function in Maple syntax:
(2*x-1)*(2*x^4-7*x^2+5*x-1)/(x-1)/(x^2+x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(2 x^{4}-7 x^{2}+5 x -1\right)}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(2*x - 1)*(2*x**4 - 7*x**2 + 5*x - 1)/((x - 1)*(x**2 - 3*x + 1)**2*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+x-1)*(x^2-3*x+1)^2*F(x)-(2*x-1)*(2*x^4-7*x^2+5*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-\left(2 x -1\right) \left(2 x^{4}-7 x^{2}+5 x -1\right) = 0
Explicit closed form in Maple syntax:
-1/20*(3+5^(1/2))*(((3*n+17/10)*5^(1/2)-7*n-9/2)*(3/2-1/2*5^(1/2))^(-n)+(-2*n+4/5*5^(1/2)-3)*(3/2+1/2*5^(1/2))^(-n)+(-7/2*5^(1/2)+15/2)*(-1/2*5^(1/2)-1/2)^(-n)-15+((1/2*5^(1/2)-1/2)^(-n)+5)*5^(1/2))
Explicit closed form in latex syntax:
-\frac{\left(3+\sqrt{5}\right) \left(\left(\left(3 n +\frac{17}{10}\right) \sqrt{5}-7 n -\frac{9}{2}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-2 n +\frac{4 \sqrt{5}}{5}-3\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\frac{7 \sqrt{5}}{2}+\frac{15}{2}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-15+\left(\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+5\right) \sqrt{5}\right)}{20}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+6) = a(n)-5*a(n+1)+4*a(n+2)+11*a(n+3)-16*a(n+4)+7*a(n+5)-1, n >= 6
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) = 20
a \! \left(5\right) = 64
a \! \left(n +6\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)+11 a \! \left(n +3\right)-16 a \! \left(n +4\right)+7 a \! \left(n +5\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21191/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[26,x]+F[9,x]
F[9,x] = F[10,x]+F[25,x]
F[10,x] = F[11,x]+F[15,x]
F[11,x] = F[0,x]*F[12,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[4,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[20,x]*F[4,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]*F[4,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[17,x]*F[23,x]*F[4,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[13,x]^2
F[25,x] = F[12,x]*F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]+F[34,x]
F[28,x] = F[29,x]*F[31,x]
F[29,x] = F[23,x]+F[30,x]
F[30,x] = F[12,x]*F[21,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[1,x]+F[31,x]
F[34,x] = F[12,x]*F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[33,x]+F[38,x]
F[38,x] = F[35,x]*F[39,x]
F[39,x] = F[1,x]+F[4,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \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_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13} \left(x \right)^{2}
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_26(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_25(x))
Eq(F_10(x), F_11(x) + F_15(x))
Eq(F_11(x), F_0(x)*F_12(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_4(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_20(x)*F_4(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_17(x)*F_23(x)*F_4(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_13(x)**2)
Eq(F_25(x), F_12(x)*F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x) + F_34(x))
Eq(F_28(x), F_29(x)*F_31(x))
Eq(F_29(x), F_23(x) + F_30(x))
Eq(F_30(x), F_12(x)*F_21(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_1(x) + F_31(x))
Eq(F_34(x), F_12(x)*F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_33(x) + F_38(x))
Eq(F_38(x), F_35(x)*F_39(x))
Eq(F_39(x), F_1(x) + F_4(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 3, 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