0132_0231_1023_1302
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 646, 2019, 6303, 19666, 61339, 191280, 596423, 1859577, 5797771, 18075955, 56355710, 175700396, 547780506, 1707811319, 5324427648, 16599913646, 51753372079, 161350916255, 503041944448, 1568328196267, 4889559062636, 15244122828265, 47526428654655, 148172606696411, 461956052120341, 1440235133424994, 4490204705493118, 13999060171700476, 43644710771358669, 136070618661045630, 424225821071209398, 1322604020121151807, 4123467519287106651, 12855687812746893450, 40080031761012703775, 124957059424665915376, 389577203759623341651, 1214580419769154339669, 3786683568365707488463, 11805700317196920036031, 36806497681455611382734, 114751199435547638538780, 357758510083156077593898, 1115379640182391928127947, 3477406425480224604653816, 10841470484429623151226510, 33800329292405574472951195, 105378902420648480033541291, 328538606216348718166055936, 1024281078044623752040170719, 3193389473836663302252551308, 9955994062761088288692168061, 31039689517935687786733990899, 96772087176461244820675139455, 301705236164727870147594038073, 940622985254319407248075614562, 2932569588900577255811081810006, 9142838872281330995409742206820, 28504524823854933031716333090665, 88868233027388379023062613899406, 277063479928662832428231441533446, 863797661944314054664468235371019, 2693052151703927842943139201264903, 8396098080970147663280449170508066, 26176419547117892303533017989283539, 81609925670087217860884907926663280, 254434337587260584604282476557841423, 793247042095962970317536916256667777, 2473097286164020741605900676092787971, 7710347296941999147167643763979653755, 24038462122803035734206548166798605310, 74944440110838637945304786688773115596, 233653429026937285772164720154367681306, 728458639697683456629764691675815090319, 2271107220468063149239931420944989611248, 7080605165178294369649700213814335500046, 22075122236992840021085612239208265252279, 68823357666477785049207693041493375785255, 214569799869573552362165856625693814128448, 668961826000737661966196056818461615622467, 2085614680715834778887042623662829731569836, 6502297182519076880465689017830604078436065, 20272138013184690455966949964904276392983255, 63202214246749714428139920151657228385822611, 197044824926412672887923487408914549804084141, 614324410829915674803169887158394578125268994, 1915272232510865380593641845887101001381736718, 5971222468062995015550271190161644976703004876, 18616412412744594584573176312022425453994726869, 58040177363181568133350014459160951128105061630, 180951201212292892053306744873039191057518017398, 564149503115454100072193879209742187918274336007, 1758842493076484172337466614272641169958611850851, 5483523247592769805078097256426567110724551956250
Generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)/(5*x^4-11*x^3+12*x^2-6*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{5 x^{4}-11 x^{3}+12 x^{2}-6 x +1}
Generating function in sympy syntax:
(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1)/(5*x**4 - 11*x**3 + 12*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(5*x^4-11*x^3+12*x^2-6*x+1)*F(x)-(2*x-1)*(x^3-2*x^2+3*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(5 x^{4}-11 x^{3}+12 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
-835/643*Sum(_alpha^(-n+2),_alpha = RootOf(5*_Z^4-11*_Z^3+12*_Z^2-6*_Z+1))+1436/643*Sum(_alpha^(-n+1),_alpha = RootOf(5*_Z^4-11*_Z^3+12*_Z^2-6*_Z+1))-1108/643*Sum(_alpha^(-n),_alpha = RootOf(5*_Z^4-11*_Z^3+12*_Z^2-6*_Z+1))+282/643*Sum(_alpha^(-n-1),_alpha = RootOf(5*_Z^4-11*_Z^3+12*_Z^2-6*_Z+1))+piecewise(n = 0,2/5,0)
Explicit closed form in latex syntax:
-\frac{835 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(5 \textit{\_Z}^{4}-11 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{643}+\frac{1436 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(5 \textit{\_Z}^{4}-11 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{643}-\frac{1108 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(5 \textit{\_Z}^{4}-11 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{643}+\frac{282 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(5 \textit{\_Z}^{4}-11 \textit{\_Z}^{3}+12 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{643}+\left(\left\{\begin{array}{cc}\frac{2}{5} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+4) = -5*a(n)+11*a(n+1)-12*a(n+2)+6*a(n+3), n >= 5
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(n +4\right) = -5 a \! \left(n \right)+11 a \! \left(n +1\right)-12 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15321/
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[24,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[12,x]+F[21,x]
F[21,x] = F[18,x]+F[19,x]+F[22,x]
F[22,x] = F[12,x]*F[14,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[18,x]+F[26,x]+F[65,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[41,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[12,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[32,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[12,x]+F[36,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[35,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[18,x]+F[26,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[2,x]+F[42,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[59,x]
F[48,x] = F[49,x]+F[57,x]
F[49,x] = F[18,x]+F[50,x]+F[56,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[12,x]+F[53,x]
F[53,x] = F[30,x]
F[54,x] = F[49,x]+F[55,x]
F[55,x] = F[46,x]
F[56,x] = F[12,x]*F[2,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[48,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[49,x]+F[60,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[59,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[2,x]+F[68,x]
F[68,x] = F[18,x]+F[50,x]+F[69,x]
F[69,x] = F[12,x]*F[67,x]
F[70,x] = F[42,x]+F[71,x]
F[71,x] = F[18,x]+F[72,x]+F[73,x]+F[77,x]
F[72,x] = 0
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[49,x]+F[75,x]
F[75,x] = F[18,x]+F[72,x]+F[73,x]+F[76,x]
F[76,x] = F[12,x]*F[42,x]
F[77,x] = F[12,x]*F[70,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_{24}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \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_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{65}\! \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_{41}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \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_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \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_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \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_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{57}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{56}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{46}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{49}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{42}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{18}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)
F_{72}\! \left(x \right) = 0
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{49}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{18}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \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_24(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_12(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_19(x) + F_22(x))
Eq(F_22(x), F_12(x)*F_14(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_18(x) + F_26(x) + F_65(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_41(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_12(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_32(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_12(x) + F_36(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_35(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_18(x) + F_26(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_2(x) + F_42(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_59(x))
Eq(F_48(x), F_49(x) + F_57(x))
Eq(F_49(x), F_18(x) + F_50(x) + F_56(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_12(x) + F_53(x))
Eq(F_53(x), F_30(x))
Eq(F_54(x), F_49(x) + F_55(x))
Eq(F_55(x), F_46(x))
Eq(F_56(x), F_12(x)*F_2(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_48(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_49(x) + F_60(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_59(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_2(x) + F_68(x))
Eq(F_68(x), F_18(x) + F_50(x) + F_69(x))
Eq(F_69(x), F_12(x)*F_67(x))
Eq(F_70(x), F_42(x) + F_71(x))
Eq(F_71(x), F_18(x) + F_72(x) + F_73(x) + F_77(x))
Eq(F_72(x), 0)
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_49(x) + F_75(x))
Eq(F_75(x), F_18(x) + F_72(x) + F_73(x) + F_76(x))
Eq(F_76(x), F_12(x)*F_42(x))
Eq(F_77(x), F_12(x)*F_70(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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