0123_0213_0321_1320_2103

Counting sequence:
1, 1, 2, 6, 19, 51, 110, 229, 507, 1128, 2474, 5413, 11893, 26145, 57414, 126058, 276854, 608068, 1335425, 2932789, 6440970, 14145667, 31066558, 68227957, 149841548, 329080592, 722723395, 1587237512, 3485874692, 7655642318, 16813242811, 36925070444, 81094459479, 178098817191, 391138788309, 859014978601, 1886559850173, 4143243316444, 9099348305937, 19983895046839, 43888424514784, 96387305973002, 211684808815719, 464899997264743, 1021008586617657, 2242328543947628, 4924578857514930, 10815309375307820, 23752471077835720, 52164932386633941, 114564087331577719, 251604468857321490, 552571143570982071, 1213551055329196542, 2665188330996188691, 5853259167370891564, 12854867508595741042, 28231727647517581685, 62002229539172868470, 136168658036990795622, 299052204564356942152, 656775372130841960811, 1442403309034216681302, 3167791293943917781753, 6957070618969880006755, 15279046851939113975697, 33555685358030321550675, 73694650638775165566443, 161847432851537272870530, 355447665381160303472169, 780630502436277986103307, 1714412670794868653566540, 3765175453186854230792218, 8269039557849297039852409, 18160379525315882839559797, 39883638504359772287799569, 87592036175729680553007378, 192368727857455963870275986, 422478219175702134557414302, 927842314423012659236663548, 2037717736345175535481451525, 4475214708867690035996779013, 9828420459443453257494076526, 21585075803446234476721163535, 47404921204083607071889816210, 104110199789364993707070921361, 228645748687539119540882073232, 502149438754853484079390194144, 1102815426436819646726804722263, 2421991883138929958081061643164, 5319153632937438925296363894772, 11681870433902107748610087638126, 25655603551182719014328379180159, 56344572326813314134501184259584, 123743369527751495380019892043475, 271763913891574009714304461472527, 596845109160402398154921947213773, 1310785082639102060047098124732161, 2878732700492720277905189855280997, 6322243112655100658785065150901760, 13884845219798806260963521558760925

Generating function in Maple syntax:
-(3*x^8+10*x^7+21*x^6+18*x^5+6*x^4+x^3+1)/(x^5+3*x^4+2*x^3+x^2+x-1)

Generating function in latex syntax:
-\frac{3 x^{8}+10 x^{7}+21 x^{6}+18 x^{5}+6 x^{4}+x^{3}+1}{x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1}

Generating function in sympy syntax:
(-3*x**8 - 10*x**7 - 21*x**6 - 18*x**5 - 6*x**4 - x**3 - 1)/(x**5 + 3*x**4 + 2*x**3 + x**2 + x - 1)

Implicit equation for the generating function in Maple syntax:
(x^5+3*x^4+2*x^3+x^2+x-1)*F(x)+3*x^8+10*x^7+21*x^6+18*x^5+6*x^4+x^3+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+3 x^{8}+10 x^{7}+21 x^{6}+18 x^{5}+6 x^{4}+x^{3}+1 = 0

Explicit closed form in Maple syntax:
1/7367*((-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+(-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2-115255*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-139179)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2-139179*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-81208)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(1-n)+1/7367*(((22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+(46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+85257*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+174563)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(1-n)+1/7367*(22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^3+68862*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+45908*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+11578)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(1-n)+1/7367*((-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(2-n)+1/7367*(22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+68862*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+53922)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(2-n)+1/7367*(22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(3-n)-11376/7367*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(1-n)+8014/7367*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(2-n)-22469/7367*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(3-n)-22954/7367*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(-n+4)+1/7367*(((22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+((22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2+(22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^2+161648*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+224436)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^2+224436*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+255771)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+(46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2+(46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^2+224436*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+255771)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+85257*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^2+255771*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+160665)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(-n)+1/7367*((((-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+(-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)-85257*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-174563)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+((-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-85257)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)-85257*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-174563)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+(-85257*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-174563)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)-174563*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-363024)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(-n)+1/7367*((-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-46393)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^3+(-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2-115255*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-139179)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+(-22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^3-115255*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2-185087*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-92786)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)-46393*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^3-139179*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2-92786*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)-9849)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(-n)+1/7367*(22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^4+68862*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^3+45908*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+22954*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+36544)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(-n)-piecewise(n = 0,-23,n = 1,12,n = 2,1,n = 3,3,0)+13590/7367*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(-n)

Explicit closed form in latex syntax:
\frac{\left(\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}-115255 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-139179\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}-139179 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-81208\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{1-n}}{7367}+\frac{\left(\left(\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+\left(46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+85257 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+174563\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{1-n}}{7367}+\frac{\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3}+68862 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+45908 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+11578\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{1-n}}{7367}+\frac{\left(\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2-n}}{7367}+\frac{\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+68862 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+53922\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2-n}}{7367}+\frac{\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{3-n}}{7367}-\frac{11376 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{1-n}}{7367}+\frac{8014 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2-n}}{7367}-\frac{22469 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3-n}}{7367}-\frac{22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{7367}+\frac{\left(\left(\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+\left(\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}+\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2}+161648 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+224436\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2}+224436 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+255771\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+\left(46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}+\left(46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2}+224436 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+255771\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+85257 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2}+255771 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+160665\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{7367}+\frac{\left(\left(\left(\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+\left(-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)-85257 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-174563\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+\left(\left(-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-85257\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)-85257 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-174563\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+\left(-85257 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-174563\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)-174563 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)-363024\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{7367}+\frac{\left(\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-46393\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3}+\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}-115255 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-139179\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+\left(-22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{3}-115255 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}-185087 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-92786\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)-46393 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{3}-139179 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}-92786 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)-9849\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{7367}+\frac{\left(22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{4}+68862 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3}+45908 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+22954 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+36544\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{7367}-\left(\left\{\begin{array}{cc}-23 & n =0 \\ 12 & n =1 \\ 1 & n =2 \\ 3 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{13590 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{7367}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 110
a(7) = 229
a(8) = 507
a(n+5) = a(n)+3*a(n+1)+2*a(n+2)+a(n+3)+a(4+n), n >= 9

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(5\right) = 51
a \! \left(6\right) = 110
a \! \left(7\right) = 229
a \! \left(8\right) = 507
a \! \left(n +5\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(4+n \right), \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16852/
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[20,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[77,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[43,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[22,x]+F[27,x]+F[34,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[33,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[4,x]
F[33,x] = F[17,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[14,x]+F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[4,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[37,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[2,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[2,x]
F[51,x] = 2*F[22,x]+F[52,x]+F[74,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[65,x]
F[54,x] = F[48,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[4,x]
F[65,x] = F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[15,x]+F[73,x]
F[73,x] = F[63,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[66,x]+F[69,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[94,x]
F[79,x] = F[2,x]+F[80,x]
F[80,x] = F[22,x]+F[81,x]+F[92,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[88,x]
F[83,x] = F[11,x]+F[84,x]
F[84,x] = F[22,x]+F[27,x]+F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[15,x]+F[87,x]
F[87,x] = F[38,x]
F[88,x] = F[48,x]+F[89,x]
F[89,x] = 2*F[22,x]+F[52,x]+F[90,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[69,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[2,x]+F[58,x]
F[94,x] = F[69,x]+F[95,x]
F[95,x] = 2*F[22,x]+F[104,x]+F[96,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[98,x]
F[98,x] = F[17,x]+F[99,x]
F[99,x] = 2*F[22,x]+F[100,x]+F[102,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[31,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[37,x]
F[104,x] = F[105,x]*F[4,x]
F[105,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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{77}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{43}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \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_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \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_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right)+F_{48}\! \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_{2}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{74}\! \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_{65}\! \left(x \right)
F_{54}\! \left(x \right) = F_{48}\! \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_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \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_{61}\! \left(x \right) = F_{4}\! \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_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \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_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{15}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{63}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{94}\! \left(x \right)
F_{79}\! \left(x \right) = F_{2}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{22}\! \left(x \right)+F_{81}\! \left(x \right)+F_{92}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \left(x \right)
F_{83}\! \left(x \right) = F_{11}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \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_{15}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{38}\! \left(x \right)
F_{88}\! \left(x \right) = F_{48}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{69}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{2}\! \left(x \right)+F_{58}\! \left(x \right)
F_{94}\! \left(x \right) = F_{69}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{104}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{17}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{100}\! \left(x \right)+F_{102}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{31}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{37}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \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_20(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_77(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_43(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_33(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_4(x))
Eq(F_33(x), F_17(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_14(x) + F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_4(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_37(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_2(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_2(x))
Eq(F_51(x), 2*F_22(x) + F_52(x) + F_74(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_65(x))
Eq(F_54(x), F_48(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_4(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_15(x) + F_73(x))
Eq(F_73(x), F_63(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_66(x) + F_69(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_94(x))
Eq(F_79(x), F_2(x) + F_80(x))
Eq(F_80(x), F_22(x) + F_81(x) + F_92(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_88(x))
Eq(F_83(x), F_11(x) + F_84(x))
Eq(F_84(x), F_22(x) + F_27(x) + F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_15(x) + F_87(x))
Eq(F_87(x), F_38(x))
Eq(F_88(x), F_48(x) + F_89(x))
Eq(F_89(x), 2*F_22(x) + F_52(x) + F_90(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_69(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_2(x) + F_58(x))
Eq(F_94(x), F_69(x) + F_95(x))
Eq(F_95(x), F_104(x) + 2*F_22(x) + F_96(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_98(x))
Eq(F_98(x), F_17(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_102(x) + 2*F_22(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_31(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_37(x))
Eq(F_104(x), F_105(x)*F_4(x))
Eq(F_105(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, 2, 1, 3], "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, 3, 2, 0], "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, 2, 1, 3], "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, 3, 2, 0], "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, 2, 1, 3], "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, 3, 2, 0], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 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, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [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": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "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": [1, 0, 2], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], 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