0123_0132_0321_1203_1302
Counting sequence:
1, 1, 2, 6, 19, 54, 146, 394, 1075, 2949, 8089, 22159, 60668, 166107, 454864, 1245663, 3411269, 9341656, 25581688, 70054347, 191841178, 525350116, 1438651994, 3939694767, 10788706393, 29544468651, 80906423585, 221559219430, 606731646025, 1661511943548, 4549988376721, 12459973169089, 34121170993538, 93439551933114, 255880721874295, 700719796598130, 1918894982577950, 5254822215757906, 14390134306475575, 39406845152148405, 107914173125227189, 295518931199248987, 809267551870171292, 2216148954831021819, 6068841112743305548, 16619294642372820063, 45511317445769762093, 124631042425149020776, 341297453199150836620, 934630323983566805487, 2559450222442506063982, 7008958807628172833656, 19193771825008430052506, 52561427022447296266539, 143937504093718665396745, 394167477147857567233983, 1079412909230006384237213, 2955931922753051228579950, 8094709130524874631199873, 22167058518308406820685832, 60703661543703607858066993, 166234709118894940697981041, 455227540038067174186952450, 1246623610120391956879658526, 3413832179792206920413177179, 9348651876294327926842139934, 25600933877617017166310279354, 70107200918247581495375832538, 191985950359754366820854011771, 525746352054749749478206964565, 1439736742094540179029529268977, 3942665276584060000339563853351, 10796841553524044889448915106012, 29566747201249035654873311185995, 80967432533751948603158246931736, 221726288877348315517372126805391, 607189157922569646562299868559525, 1662764823086269793840396224018072, 4553419343574127117567858996241968, 12469368746899034454232868218989267, 34146900431115802735498690532573650, 93510011029428247288379953375774684, 256073671470217970616685296488411450, 701248181861518764303501190427814279, 1920341945896914961565363915085882985, 5258784670759250351763572157001307379, 14400985341439291379769779157399708473, 39436560305939874400209527398875231766, 107995546963635438306305365161165682921, 295741770415461882133451148521800831140, 809877788738109875631122084877274238113, 2217820065694172428102893805786231051729, 6073417387405683426695865230608046792066, 16631826599556137512620626016370764010290, 45545635742295470622912424071943219694719, 124725021797979991187801798849260202709306, 341554812200378672685720550527629584145126, 935335091992947071381447664937752919031074, 2561380203304554219958686900565477021920319, 7014243988110680734235280854460978812120997, 19208245094294138804171398326445911196442509
Generating function in Maple syntax:
-(x-1)^3/(x^5-x^4-4*x^3+5*x^2-4*x+1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{3}}{x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1}
Generating function in sympy syntax:
-(x - 1)**3/(x**5 - x**4 - 4*x**3 + 5*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-x^4-4*x^3+5*x^2-4*x+1)*F(x)+(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-1586/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+3)-1586/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+3)-1586/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+3)-1586/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+3)-1586/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+3)-1308/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+2)-1308/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+2)-1308/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+2)-1308/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+2)-1308/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+2)+2643/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n+1)+2643/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n+1)+2643/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n+1)+2643/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n+1)+2643/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n+1)+5911/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n-1)+5911/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n-1)+5911/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n-1)+5911/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n-1)+5911/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n-1)+15412/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 1)^(-n)+15412/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 2)^(-n)+15412/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 3)^(-n)+15412/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 4)^(-n)+15412/94747*RootOf(_Z^5-_Z^4-4*_Z^3+5*_Z^2-4*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{1586 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{94747}-\frac{1586 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{94747}-\frac{1586 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{94747}-\frac{1586 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{94747}-\frac{1586 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{94747}-\frac{1308 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{94747}-\frac{1308 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{94747}-\frac{1308 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{94747}-\frac{1308 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{94747}-\frac{1308 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{94747}+\frac{2643 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{94747}+\frac{2643 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{94747}+\frac{2643 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{94747}+\frac{2643 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{94747}+\frac{2643 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{94747}+\frac{5911 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{94747}+\frac{5911 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{94747}+\frac{5911 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{94747}+\frac{5911 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{94747}+\frac{5911 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{94747}+\frac{15412 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{94747}+\frac{15412 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{94747}+\frac{15412 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{94747}+\frac{15412 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{94747}+\frac{15412 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+5 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{94747}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = -a(n)+a(n+1)+4*a(n+2)-5*a(n+3)+4*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) = -a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-5 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16232/
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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[66,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[7,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]+F[31,x]
F[31,x] = F[18,x]
F[32,x] = F[20,x]+F[33,x]
F[33,x] = F[15,x]+F[34,x]+F[57,x]+F[63,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[31,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[46,x]+F[47,x]+F[56,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[37,x]
F[45,x] = F[41,x]
F[46,x] = 0
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[2,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[2,x]*F[4,x]
F[56,x] = 0
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[54,x]
F[60,x] = F[61,x]
F[61,x] = 2*F[15,x]+F[50,x]+F[62,x]
F[62,x] = F[4,x]*F[59,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[49,x]+F[51,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[53,x]+F[68,x]
F[68,x] = F[51,x]+F[61,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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{66}\! \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_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \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_{18}\! \left(x \right)
F_{32}\! \left(x \right) = F_{20}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \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_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{56}\! \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_{37}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = 0
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{54}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{53}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{51}\! \left(x \right)+F_{61}\! \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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_66(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_7(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x) + F_31(x))
Eq(F_31(x), F_18(x))
Eq(F_32(x), F_20(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_57(x) + F_63(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_31(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_46(x) + F_47(x) + F_56(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_37(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), 0)
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_2(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_2(x)*F_4(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_54(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), 2*F_15(x) + F_50(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_59(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_49(x) + F_51(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_53(x) + F_68(x))
Eq(F_68(x), F_51(x) + F_61(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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": "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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": []}, "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [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, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 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2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [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]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [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, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [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, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [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], "pos": [[0, 0]]}], "requirements": []}, {"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]]}, {"patt": [2, 1, 0], "pos": [[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, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": 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