0123_0132_1023_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 56, 160, 456, 1305, 3743, 10739, 30805, 88354, 253411, 726828, 2084693, 5979343, 17150020, 49189864, 141086869, 404666824, 1160669646, 3329044948, 9548401919, 27386827325, 78551187667, 225301347009, 646211705844, 1853471247987, 5316145832920, 15247825693425, 43733974891825, 125438249249570, 359783312944906, 1031934302719655, 2959804879257772, 8489343653167836, 24349225236650888, 69838705304848917, 200312113065347415, 574537320896167687, 1647894019244166057, 4726507051665422586, 13556617506075745347, 38883233685493479872, 111525302027817470669, 319878050601440231751, 917477607288215682116, 2631518975098816410020, 7547750551398193553265, 21648537945272655405772, 62092565457279530917242, 178094552842930324885372, 510812679726455483150251, 1465118329584442959659429, 4202267885820328040678583, 12052989187027340038299133, 34570511040668417491102580, 99155505333007981467231403, 284398868916867393677133732, 815715843205615625474776617, 2339644807276463769192500801, 6710593976824253387777656898, 19247396605560367181585341246, 55205586475827241008031411771, 158341246901911092893998216240, 454156038744922558569089176008, 1302615152805078621784507965704, 3736174555790531222362613623633, 10716135369127996914041478408799, 30736132783597461777656445240667, 88157701069411597171203394357021, 252854850431645584653082650757522, 725240955823810857202958064719379, 2080143778560505736211610081620580, 5966290382165308810626365510054645, 17112577164714894527661205846153295, 49082474780927934544055642273818372, 140778873189704241912129231449134032, 403783452750100780091118668458747149, 1158135968989392060273979407477311136, 3321777832974021788514345024445257718, 9527558306704017029926361357163511620, 27327043484534753939245545903762347959, 78379715092396170567446692063780387709, 224809527654974117593764957392384344603, 644801064470257686457769474468198682041, 1849425231567929726944309596966511608436, 5304541005945966170176044149569969569843, 15214540606168172695343702477407001953952, 43638506252900506555330717383021355490385, 125164425090324503135100383447367670266849, 358997927596345673780853729693606526348162, 1029681652158475278119754094410672461454690, 2953343803098321167216524812850880758690959, 8470811926205752137563951868917252878658628, 24296072341416048338755030911654911225611748, 69686251608672984970881226930243932136784456, 199874843761855779246293565569811662469379725, 573283140455988912024951196382417308210252919, 1644296765643308683365811884285567417134004527
Generating function in Maple syntax:
-(2*x-1)*(x-1)^2/(2*x^5+x^4-6*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{2 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**2/(2*x**5 + x**4 - 6*x**3 + 8*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^5+x^4-6*x^3+8*x^2-5*x+1)*F(x)+(2*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-17884/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+3)-17884/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+3)-17884/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+3)-17884/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+3)-17884/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+3)-24264/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+2)-24264/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+2)-24264/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+2)-24264/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+2)-24264/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+2)+34887/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+1)+34887/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+1)+34887/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+1)+34887/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+1)+34887/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+1)+13359/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n-1)+13359/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n-1)+13359/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n-1)+13359/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n-1)+13359/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n-1)-20732/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n)-20732/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n)-20732/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n)-20732/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n)-20732/91363*RootOf(2*_Z^5+_Z^4-6*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{17884 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{91363}-\frac{17884 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{91363}-\frac{17884 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{91363}-\frac{17884 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{91363}-\frac{17884 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{91363}-\frac{24264 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{91363}-\frac{24264 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{91363}-\frac{24264 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{91363}-\frac{24264 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{91363}-\frac{24264 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{91363}+\frac{34887 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{91363}+\frac{34887 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{91363}+\frac{34887 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{91363}+\frac{34887 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{91363}+\frac{34887 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{91363}+\frac{13359 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{91363}+\frac{13359 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{91363}+\frac{13359 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{91363}+\frac{13359 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{91363}+\frac{13359 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{91363}-\frac{20732 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{91363}-\frac{20732 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{91363}-\frac{20732 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{91363}-\frac{20732 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{91363}-\frac{20732 \mathit{RootOf}\! \left(2 \textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{91363}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = -2*a(n)-a(n+1)+6*a(n+2)-8*a(n+3)+5*a(n+4), 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) = 19
a \! \left(n +5\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16312/
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[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[53,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[1,x]+F[24,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[28,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[13,x]+F[21,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[48,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]+F[42,x]+F[47,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[31,x]+F[4,x]
F[45,x] = F[41,x]+F[46,x]
F[46,x] = F[38,x]
F[47,x] = F[2,x]*F[4,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[41,x]
F[52,x] = F[49,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[2,x]+F[41,x]
F[56,x] = F[57,x]+F[69,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[2,x]+F[61,x]
F[61,x] = F[47,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = 2*F[13,x]+F[64,x]+F[68,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[61,x]
F[67,x] = F[63,x]
F[68,x] = F[4,x]*F[57,x]
F[69,x] = 2*F[13,x]+F[68,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[51,x]+F[72,x]
F[72,x] = F[69,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_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{31}\! \left(x \right)+F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{38}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{41}\! \left(x \right)
F_{52}\! \left(x \right) = F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{69}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{2}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{47}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{63}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{51}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \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_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_53(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_1(x) + F_24(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_13(x) + F_21(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_48(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x) + F_42(x) + F_47(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_31(x) + F_4(x))
Eq(F_45(x), F_41(x) + F_46(x))
Eq(F_46(x), F_38(x))
Eq(F_47(x), F_2(x)*F_4(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_41(x))
Eq(F_52(x), F_49(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_2(x) + F_41(x))
Eq(F_56(x), F_57(x) + F_69(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_2(x) + F_61(x))
Eq(F_61(x), F_47(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), 2*F_13(x) + F_64(x) + F_68(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_61(x))
Eq(F_67(x), F_63(x))
Eq(F_68(x), F_4(x)*F_57(x))
Eq(F_69(x), 2*F_13(x) + F_68(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_51(x) + F_72(x))
Eq(F_72(x), F_69(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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]]}, {"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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"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, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": 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[1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "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, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": 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"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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