0132_0213_1023_1302
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 644, 2001, 6202, 19205, 59451, 184019, 569583, 1762997, 5456928, 16890666, 52281333, 161825653, 500897181, 1550422638, 4799011421, 14854346913, 45978563202, 142317157151, 440513417944, 1363518493706, 4220490524046, 13063658815122, 40435864349254, 125160887349219, 387409741174591, 1199147040592736, 3711712620265360, 11488841745119574, 35561342745180271, 110072810295776616, 340707707628066782, 1054590518110793755, 3264267687530748320, 10103868139284755651, 31274442278867355081, 96803593076739035217, 299635579398021793881, 927460206668959513715, 2870761999235534160330, 8885852349240652754536, 27504325330188080604539, 85133972762165446009151, 263514673828934280578807, 815655384920919379593620, 2524693206961179295099751, 7814667698042545643946139, 24188693922274586729710948, 74871118808038579164471477, 231748123713506554739191398, 717328573417343844353074216, 2220342818727954480059933588, 6872613771943747913345694068, 21272760071064519241108060156, 65845446297077452569819210533, 203811013877728101882507625261, 630855005378119541906968557118, 1952681703695293617480905094122, 6044123932504828288532533338376, 18708340454230107839659743752349, 57907813681499689009010487288546, 179241707385813091624574879022696, 554805778772572152357344004720517, 1717286990002212319878930493218694, 5315508091057128374259598925058849, 16453060222658181043648986545004711, 50927058345720707125044683179390911, 157634217382656661683208427120144915, 487924244930016194040033635449437889, 1510268980576799181144566601071324596, 4674726491649619448607675875839386486, 14469652792169958817854188199889129697, 44787829255882093591664919379858767385, 138631498506967514232344034301030383265, 429105243491194629060338331101838698426, 1328206879206338018725481637902900322161, 4111190764340404412857043122283328187557, 12725344044970961466164669818300269388006, 39388680882303575482741428688903216513035, 121919547020898446635444606172617366559220, 377376840575011184693631501851950861905366, 1168092264794636205121654542042350379559434, 3615588961405417492929225127788949942877926, 11191310765280168680551642992863264803592162, 34640396898543349629876495699434575222598535, 107222212165830372271194275082035637670999051, 331884268399297181069641626341359686962281884, 1027279379766783840401594124207447873589007188, 3179731685336679465219997358688515821393943082, 9842204360248524655961616323279927503106671243, 30464515957621843669557739090609902274131979516, 94296633006384854252420591637577394225608877218, 291875800971529888709976521002916073304646451983, 903441411179588842691320671512395928672685667756, 2796416766026386724866301970635093105809681606303
Generating function in Maple syntax:
(2*x-1)^3/(x^4-6*x^2+5*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{\left(2 x -1\right)^{3}}{\left(x^{4}-6 x^{2}+5 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x - 1)**3/((x - 1)**2*(x**4 - 6*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^4-6*x^2+5*x-1)*(x-1)^2*F(x)-(2*x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-6 x^{2}+5 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(2 x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
1/580*((-65*5^(1/2)+155)*(7+2*5^(1/2))^(1/2)+87*5^(1/2)-145)*(-1/2*5^(1/2)-1/2*(7+2*5^(1/2))^(1/2))^(-n)+1/580*((65*5^(1/2)-155)*(7+2*5^(1/2))^(1/2)+87*5^(1/2)-145)*(-1/2*5^(1/2)+1/2*(7+2*5^(1/2))^(1/2))^(-n)+1/580*((65*5^(1/2)+155)*(7-2*5^(1/2))^(1/2)-87*5^(1/2)-145)*(1/2*5^(1/2)-1/2*(7-2*5^(1/2))^(1/2))^(-n)+1/580*((-65*5^(1/2)-155)*(7-2*5^(1/2))^(1/2)-87*5^(1/2)-145)*(1/2*5^(1/2)+1/2*(7-2*5^(1/2))^(1/2))^(-n)-n+2
Explicit closed form in latex syntax:
\frac{\left(\left(-65 \sqrt{5}+155\right) \sqrt{7+2 \sqrt{5}}+87 \sqrt{5}-145\right) \left(-\frac{\sqrt{5}}{2}-\frac{\sqrt{7+2 \sqrt{5}}}{2}\right)^{-n}}{580}+\frac{\left(\left(65 \sqrt{5}-155\right) \sqrt{7+2 \sqrt{5}}+87 \sqrt{5}-145\right) \left(-\frac{\sqrt{5}}{2}+\frac{\sqrt{7+2 \sqrt{5}}}{2}\right)^{-n}}{580}+\frac{\left(\left(65 \sqrt{5}+155\right) \sqrt{7-2 \sqrt{5}}-87 \sqrt{5}-145\right) \left(\frac{\sqrt{5}}{2}-\frac{\sqrt{7-2 \sqrt{5}}}{2}\right)^{-n}}{580}+\frac{\left(\left(-65 \sqrt{5}-155\right) \sqrt{7-2 \sqrt{5}}-87 \sqrt{5}-145\right) \left(\frac{\sqrt{5}}{2}+\frac{\sqrt{7-2 \sqrt{5}}}{2}\right)^{-n}}{580}-n +2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n) = 6*a(n+2)-5*a(n+3)+a(n+4)+n-1, n >= 4
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(n \right) = 6 a \! \left(n +2\right)-5 a \! \left(n +3\right)+a \! \left(n +4\right)+n -1, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15258/
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[12,x]*F[4,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[49,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[31,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[1,x]+F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[28,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[18,x]+F[19,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[2,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]+F[47,x]
F[39,x] = F[18,x]+F[40,x]+F[46,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[12,x]+F[43,x]
F[43,x] = F[26,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = F[36,x]
F[46,x] = F[12,x]*F[2,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[38,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[2,x]+F[52,x]
F[52,x] = F[18,x]+F[40,x]+F[53,x]
F[53,x] = F[12,x]*F[51,x]
F[54,x] = F[17,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[54,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{47}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{26}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{36}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_16(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_49(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_31(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_28(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_18(x) + F_19(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_2(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x) + F_47(x))
Eq(F_39(x), F_18(x) + F_40(x) + F_46(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_12(x) + F_43(x))
Eq(F_43(x), F_26(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), F_36(x))
Eq(F_46(x), F_12(x)*F_2(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_38(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_2(x) + F_52(x))
Eq(F_52(x), F_18(x) + F_40(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_51(x))
Eq(F_54(x), F_17(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_54(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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