0132_1203_1302_2130
Counting sequence:
1, 1, 2, 6, 20, 64, 195, 572, 1632, 4563, 12568, 34228, 92419, 247892, 661496, 1758099, 4657856, 12309676, 32467779, 85503260, 224893968, 590948115, 1551620392, 4071515236, 10678653955, 27996952484, 73379312360, 192279396627, 503744090192, 1319540076508, 3456084098883, 9051195248108, 23702601919104, 62067079491987, 162520111393336, 425537278102804, 1114181917228291, 2917193157175796, 7637775511421144, 19996906471200915, 52354524450146528, 137069896694645260, 358861762703555907, 939528860433462716, 2459752306387526640, 6439784133822133779, 16859714444288163400, 44139592295507444932, 115559537431257370627, 302539987568497109828, 792062395599070933448, 2073651210247133817363, 5428899397916655127664, 14213063590526457494332, 37210325150659922634051, 97417980541347627807884, 255043756084971409274976, 667713571440343124358291, 1748097534696810267223384, 4576580203585990793640820, 11981645453961765365320963, 31368360986158105843493780, 82123447304345341323359672, 215002000814773872594695571, 562882595492228937700371200, 1473645867519339767592553324, 3858055173086487028463253315, 10100519988393200662996523612, 26443505474622645687779727312, 69229997818980541928559029523, 181246490786224079301749206888, 474509480221288870679230488868, 1242281961388410834730702468099, 3252336427260628142681750158052, 8514727367617138422011000143016, 22291845771218708401461565848339, 58360810139656012580029151163536, 152790585039705747416806440379036, 400010945772818798792490565874499, 1047242253884355253136344629901100, 2741715819129235100830859230851648, 7177905210076884193510779200118675, 18791999824399601495462399291272312, 49198094290020519779301917810910676, 128802283100063619725101667003237635, 337208755120182588980934338097064244, 882823982482926498026793115433891480, 2311263192778317309996087061195307667, 6050965596761137648311669207531075872, 15841633599342662880753156906951307468, 41473935204980671111803942338020219203, 108580172023104361942826288023685290748, 284266580879497180197920830100549924784, 744219570646026194623249364081717808147, 1948392131120478405636096275889559710856, 5100956822840440966252862871351772871044, 13354478337653384261136709914245180243971, 34962478190629743113342843544642207041156, 91532956235265811191577257108397512233864, 239636390517247429727388366642380858356755, 627376215320675570603843852711202892204208
Generating function in Maple syntax:
-(x^6-2*x^5+7*x^4-19*x^3+18*x^2-7*x+1)/(x-1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{x^{6}-2 x^{5}+7 x^{4}-19 x^{3}+18 x^{2}-7 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**6 + 2*x**5 - 7*x**4 + 19*x**3 - 18*x**2 + 7*x - 1)/((x - 1)*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)*(2*x-1)^2*F(x)+x^6-2*x^5+7*x^4-19*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{6}-2 x^{5}+7 x^{4}-19 x^{3}+18 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(-n)+(3/2+1/2*5^(1/2))^(-n)+1+1/8*(-n-10)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+1+\frac{\left(-n -10\right) 2^{n}}{8} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(6) = 195
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1, 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 195
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15553/
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[27,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[26,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[10,x]
F[25,x] = F[20,x]
F[26,x] = F[12,x]*F[13,x]
F[27,x] = F[28,x]+F[42,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[1,x]+F[14,x]
F[32,x] = F[28,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[28,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[21,x]+F[43,x]+F[47,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[6,x]
F[46,x] = F[42,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[56,x]
F[49,x] = F[28,x]+F[50,x]
F[50,x] = F[21,x]+F[51,x]+F[55,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[10,x]
F[54,x] = F[50,x]
F[55,x] = F[12,x]*F[49,x]
F[56,x] = F[57,x]+F[67,x]
F[57,x] = F[21,x]+F[58,x]+F[62,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[14,x]
F[61,x] = F[33,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[36,x]+F[64,x]
F[64,x] = F[57,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[64,x]
F[67,x] = F[21,x]+F[68,x]+F[69,x]+F[73,x]
F[68,x] = 0
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[50,x]
F[72,x] = F[67,x]
F[73,x] = F[12,x]*F[56,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_{27}\! \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_{20}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \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_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{42}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{21}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{6}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{28}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{10}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{57}\! \left(x \right) = F_{21}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right)
F_{61}\! \left(x \right) = F_{33}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{36}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{57}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{21}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{68}\! \left(x \right) = 0
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{50}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \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_27(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_26(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_10(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), F_12(x)*F_13(x))
Eq(F_27(x), F_28(x) + F_42(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_1(x) + F_14(x))
Eq(F_32(x), F_28(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_28(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_21(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_6(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_56(x))
Eq(F_49(x), F_28(x) + F_50(x))
Eq(F_50(x), F_21(x) + F_51(x) + F_55(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_10(x))
Eq(F_54(x), F_50(x))
Eq(F_55(x), F_12(x)*F_49(x))
Eq(F_56(x), F_57(x) + F_67(x))
Eq(F_57(x), F_21(x) + F_58(x) + F_62(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_14(x))
Eq(F_61(x), F_33(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_36(x) + F_64(x))
Eq(F_64(x), F_57(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_64(x))
Eq(F_67(x), F_21(x) + F_68(x) + F_69(x) + F_73(x))
Eq(F_68(x), 0)
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_50(x))
Eq(F_72(x), F_67(x))
Eq(F_73(x), F_12(x)*F_56(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:
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