0123_0132_0231_2103
Counting sequence:
1, 1, 2, 6, 20, 62, 181, 531, 1570, 4651, 13772, 40755, 120596, 356876, 1056127, 3125466, 9249340, 27371962, 81003041, 239715936, 709402175, 2099365721, 6212747107, 18385661085, 54409511491, 161016507976, 476503374527, 1410137809551, 4173084070610, 12349594871612, 36546710038801, 108154318303191, 320065925364205, 947185449331366, 2803048385748692, 8295186817319490, 24548318425068686, 72646939818201551, 214987347547189327, 636221700749406636, 1882799416442963455, 5571852765132631012, 16489033810605399461, 48796737363324065796, 144406373633228864660, 427348258770536035211, 1264670870678144738152, 3742597233795230910720, 11075635866350610327556, 32776626011556464045851, 96997339535632538745902, 287048577656330965901721, 849475731282853300634375, 2513891634406511846700287, 7439472273086944881498271, 22015963991660145053111001, 65152829755887095092083040, 192809691495117439802934631, 570590368426526288243273237, 1688573670838343571115256400, 4997071804263432945196784726, 14788058731584466275934606001, 43762965515566757442305457127, 129509706816770359497614541681, 383263884477838664684463387370, 1134210004450577227409680982062, 3356518540609227026919524347856, 9933095872233082881990480537306, 29395456158888192302163975326345, 86991291929900500580808336198251, 257437232160284756584021634862322, 761845548353889763283514210658155, 2254563703455554521322434037751196, 6672031494995442225575150473422751, 19744842073870761555692164496637752, 58431796794502894516020173534841516, 172919837183825460741523590454733315, 511729430413369337082910350987993962, 1514383856797234318729020557420335216, 4481584074372108835319144868260387586, 13262552770565423482572434531402797825, 39248467299294470180131640616071187196, 116149749749731896136955481482806389407, 343727164274969138313220058733698981985, 1017207214953851740514039803971776221091, 3010266937548296398460020970621001265601, 8908417972347460191204668759626839895027, 26363080888327363542545888138153046637476, 78017447776010518596532826613281562936835, 230880532638259025972837629486014253927987, 683255116270505932312628490184613583371726, 2021987512655552959477353346770181185852216, 5983758341468968686906867962606445505100105, 17708004458482000820545851854467935345655091, 52404091877921401666723470019753816122851617, 155081779654410694257907857688358397461211390, 458940466649169298448805477776805008848006036, 1358163108506517805384267264730311170817760926, 4019273006749634972974359321448790029677316202, 11894414891411936608053408219208352911048655811, 35199675506355769039843481976439263614090110055
Generating function in Maple syntax:
(x^4-x^3-2*x+1)/(2*x^6-3*x^5-x^4-2*x^3+x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{4}-x^{3}-2 x +1}{2 x^{6}-3 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1}
Generating function in sympy syntax:
(x**4 - x**3 - 2*x + 1)/(2*x**6 - 3*x**5 - x**4 - 2*x**3 + x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^6-3*x^5-x^4-2*x^3+x^2-3*x+1)*F(x)-x^4+x^3+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{6}-3 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+2 x -1 = 0
Explicit closed form in Maple syntax:
858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+4)+858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+4)+858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+4)+858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+4)+858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+4)+858708/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+4)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+3)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+3)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+3)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+3)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+3)-11085262/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+3)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+2)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+2)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+2)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+2)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+2)+3896392/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+2)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+1)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+1)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+1)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+1)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+1)+377073/5196749*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n-1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n-1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n-1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n-1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n-1)+960707/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n-1)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n)+33715409/150705721*RootOf(2*_Z^6-3*_Z^5-_Z^4-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{150705721}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{5196749}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{150705721}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(n+6) = -2*a(n)+3*a(n+1)+a(n+2)+2*a(n+3)-a(n+4)+3*a(n+5), n >= 6
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) = 62
a \! \left(n +6\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14798/
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[15,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[13,x] = F[14,x]*F[4,x]
F[14,x] = F[4,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[18,x]+F[44,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[25,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[13,x]+F[17,x]+F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[21,x]
F[25,x] = F[26,x]+F[36,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]*F[4,x]
F[32,x] = F[26,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[30,x]
F[36,x] = 2*F[17,x]+F[37,x]+F[39,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[30,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[16,x]+F[42,x]
F[42,x] = F[37,x]
F[43,x] = F[36,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[56,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[17,x]+F[31,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[4,x]+F[51,x]
F[51,x] = F[13,x]+F[17,x]+F[52,x]
F[52,x] = F[4,x]*F[7,x]
F[53,x] = F[30,x]+F[54,x]
F[54,x] = 2*F[17,x]+F[37,x]+F[55,x]
F[55,x] = F[16,x]*F[4,x]
F[56,x] = F[16,x]+F[57,x]
F[57,x] = 2*F[17,x]+F[37,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[12,x]
F[61,x] = F[33,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_{15}\! \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_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{44}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \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_{30}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{36}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{31}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{53}\! \left(x \right) = F_{30}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{37}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{37}\! \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_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right)
F_{61}\! \left(x \right) = F_{33}\! \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_15(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))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_4(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_44(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_25(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_13(x) + F_17(x) + F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_21(x))
Eq(F_25(x), F_26(x) + F_36(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x)*F_4(x))
Eq(F_32(x), F_26(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_30(x))
Eq(F_36(x), 2*F_17(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_30(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_16(x) + F_42(x))
Eq(F_42(x), F_37(x))
Eq(F_43(x), F_36(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_56(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_17(x) + F_31(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_4(x) + F_51(x))
Eq(F_51(x), F_13(x) + F_17(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_7(x))
Eq(F_53(x), F_30(x) + F_54(x))
Eq(F_54(x), 2*F_17(x) + F_37(x) + F_55(x))
Eq(F_55(x), F_16(x)*F_4(x))
Eq(F_56(x), F_16(x) + F_57(x))
Eq(F_57(x), 2*F_17(x) + F_37(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_12(x))
Eq(F_61(x), F_33(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 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, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "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": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [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, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": 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