0132_1203_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 647, 2029, 6363, 19952, 62545, 196016, 614210, 1924402, 6028997, 18887512, 59168595, 185352679, 580631379, 1818855238, 5697614857, 17847864050, 55908547382, 175133554956, 548605274881, 1718502759126, 5383196669207, 16862815852493, 52822607995587, 165466282311744, 518321388344825, 1623636173392096, 5086022591772754, 15931909719795274, 49906530497990085, 156331650912488032, 489707150258720579, 1534002167747894535, 4805244601221041595, 15052374822367247734, 47151395222085464721, 147701216225925922530, 462672401355291095526, 1449316101210822952516, 4539966408658675899265, 14221393779356053430878, 44548356261957213687847, 139547225552403138683309, 437130116389795422043755, 1369305178859459132133328, 4289333089806655835968289, 13436287716502900591660048, 42089020325017366431602146, 131843383327090861174250402, 412997917099825675854458245, 1293711335553184537497265640, 4052536224604369074467613363, 12694524195902908439453941031, 39765454428690396769879204323, 124564839257938751015428272614, 390197959567405720117626563993, 1222290724714233644291415847378, 3828811963492367705627009865238, 11993710461319835450268606903228, 37570163278205100080064736646081, 117688114391543649732850506340710, 368656696178668329020964516062647, 1154812959150706909289392880992141, 3617438620933265450571042729606099, 11331585840397187083004142317178528, 35496065341714408260631826257026505, 111191025906669106970255519106217280, 348304639490511938277635964517947890, 1091060370217688967401204597462268218, 3417734352321153172279804906987288581, 10706014462522813285091778370258298864, 33536470028428111084632555507957253027, 105052615602634635371055060832843832391, 329076138174348568362383942916736243659, 1030827305865072253835356900582271610518, 3229054954917642433586244272015118628641, 10114978369852153959249816771430574991810, 31685056108060704020082805140353406138822, 99253082296569363412593948917193002691956, 310909165247254322456368927244511371339201, 973919467265609098205652525888365051240942, 3050791789829052983058015280149744435451847, 9556571007888113458708332566530941914401645, 29935851320068314582375432124285404171487995, 93773717949412830261026893376716463501188784, 293745117987042531608751917344887438005290993, 920153282050408706918528240229229189608661616, 2882369818672142876268500915210446727687423618, 9028991075355358053600760122508946734759451410, 28283213108442403464844059546395929794440828741, 88596847317858778474863878356050442559284662904, 277528628892626473485342799703260549087998166035, 869355312144335883699752937693445590900614919271, 2723245748625019071115553397169670458684125572979, 8530536713592862942387064783012953429449762295174
Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2/(3*x^6-9*x^5+20*x^4-24*x^3+18*x^2-7*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{3 x^{6}-9 x^{5}+20 x^{4}-24 x^{3}+18 x^{2}-7 x +1}
Generating function in sympy syntax:
(x**3 - 2*x**2 + 3*x - 1)**2/(3*x**6 - 9*x**5 + 20*x**4 - 24*x**3 + 18*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(3*x^6-9*x^5+20*x^4-24*x^3+18*x^2-7*x+1)*F(x)-(x^3-2*x^2+3*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{6}-9 x^{5}+20 x^{4}-24 x^{3}+18 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1/3,0)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+4)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+4)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+4)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+4)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+4)-128805/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+4)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+3)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+3)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+3)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+3)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+3)+332463/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+3)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+2)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+2)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+2)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+2)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+2)-99195/27742*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+2)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n+1)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n+1)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n+1)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n+1)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n+1)+690667/194194*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n+1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n-1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n-1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n-1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n-1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n-1)+43107/97097*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n-1)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 1)^(-n)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 2)^(-n)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 3)^(-n)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 4)^(-n)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 5)^(-n)-29737/14938*RootOf(3*_Z^6-9*_Z^5+20*_Z^4-24*_Z^3+18*_Z^2-7*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{194194}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{27742}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{194194}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{97097}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf}\! \left(3 \textit{\_Z}^{6}-9 \textit{\_Z}^{5}+20 \textit{\_Z}^{4}-24 \textit{\_Z}^{3}+18 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{14938}+\left(\left\{\begin{array}{cc}\frac{1}{3} & 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(5) = 65
a(6) = 206
a(n+6) = -3*a(n)+9*a(n+1)-20*a(n+2)+24*a(n+3)-18*a(n+4)+7*a(n+5), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 206
a \! \left(n +6\right) = -3 a \! \left(n \right)+9 a \! \left(n +1\right)-20 a \! \left(n +2\right)+24 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15552/
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[29,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[23,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[22,x]+F[7,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[22,x]+F[28,x]
F[28,x] = F[26,x]
F[29,x] = F[2,x]+F[30,x]
F[30,x] = F[15,x]+F[31,x]+F[52,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[50,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]+F[16,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[35,x]
F[39,x] = 2*F[15,x]+F[40,x]+F[44,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[27,x]+F[42,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[38,x]+F[43,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[43,x]+F[49,x]
F[49,x] = F[47,x]
F[50,x] = F[51,x]+F[72,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[2,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[54,x]
F[57,x] = F[51,x]+F[58,x]
F[58,x] = 2*F[15,x]+F[59,x]+F[66,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[55,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[61,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[51,x]+F[65,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[65,x]+F[71,x]
F[71,x] = F[69,x]
F[72,x] = 2*F[15,x]+F[59,x]+F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[67,x]+F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[73,x]
F[77,x] = 3*F[15,x]+F[78,x]+F[82,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[70,x]+F[80,x]
F[80,x] = F[81,x]+F[84,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[76,x]+F[81,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[81,x]+F[87,x]
F[87,x] = F[85,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_{29}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \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_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{50}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \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_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{72}\! \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_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{51}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{51}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{65}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{59}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{67}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{78}\! \left(x \right)+F_{82}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{70}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{76}\! \left(x \right)+F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{81}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{85}\! \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_29(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_22(x) + F_7(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_22(x) + F_28(x))
Eq(F_28(x), F_26(x))
Eq(F_29(x), F_2(x) + F_30(x))
Eq(F_30(x), F_15(x) + F_31(x) + F_52(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_50(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_35(x))
Eq(F_39(x), 2*F_15(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_27(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_38(x) + F_43(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_43(x) + F_49(x))
Eq(F_49(x), F_47(x))
Eq(F_50(x), F_51(x) + F_72(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_2(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_54(x))
Eq(F_57(x), F_51(x) + F_58(x))
Eq(F_58(x), 2*F_15(x) + F_59(x) + F_66(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_55(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_61(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_51(x) + F_65(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_65(x) + F_71(x))
Eq(F_71(x), F_69(x))
Eq(F_72(x), 2*F_15(x) + F_59(x) + F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_67(x) + F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_73(x))
Eq(F_77(x), 3*F_15(x) + F_78(x) + F_82(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_70(x) + F_80(x))
Eq(F_80(x), F_81(x) + F_84(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_76(x) + F_81(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_81(x) + F_87(x))
Eq(F_87(x), F_85(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, 3, 2], "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]]}, {"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, 3, 2], "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]]}, {"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, 3, 2], "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]]}, {"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, 2, 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