0132_0231_0312_1032_2103
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 430, 1194, 3337, 9341, 26128, 73028, 204071, 570303, 1593929, 4454969, 12451380, 34800528, 97264105, 271843916, 759778780, 2123512927, 5935026211, 16587858244, 46361551761, 129576311443, 362153985522, 1012187403357, 2828971599655, 7906717937521, 22098556413355, 61763452216627, 172623222941942, 482466184010142, 1348449036864504, 3768792228858898, 10533430983157010, 29439980115406524, 82282062756505568, 229970870392458763, 642747634871383395, 1796421092061116569, 5020833317647603922, 14032771778846102147, 39220318887113746340, 109617218739761628258, 306370141426585620414, 856276638258647205332, 2393215206329839580850, 6688818505496033151017, 18694638443351076076130, 52249811568433701597743, 146033463937238859608902, 408150229624801826459858, 1140742713700010336629027, 3188271975384890888885895, 8910929753874247517102988, 24905236972104790078088684, 69607868737492545123923682, 194547652592221451919966489, 543742967794061800490359511, 1519712065841267733184766564, 4247456794582853857727590751, 11871254843173958707205668563, 33179075942883533080779647001, 92732495002971707730661014728, 259178876599202294721875526842, 724381351683375726937529518385, 2024579894595659668716905015152, 5658516387915789489008371024531, 15815037874169056234918743553448, 44201590278247319957425494862262, 123539418537668364433870762571190, 345281421698892906536270171218348, 965030122220120430195541827857225, 2697171287728066592249715290390498, 7538348065870354838091520019812549, 21068996181580546164583720332230277, 58885925168301973961005623638717871, 164580796970211056013851974819158736, 459988335989166982921896722803340291, 1285625499093798889432729675660525857, 3593206163295203104467673052998837896, 10042683923929110866770866723455872560, 28068386786761260640145647110995941639, 78448584340489822044175424886767238421, 219256647408379969148901772719354790182, 612802357581236762167575030638833592586, 1712726769727891805344165203493800416973, 4786915310379930156912861566566405624006, 13378992255951202595460910449521896201034, 37393064672914695123880594367008403459788, 104510209654304833346684543882690195873996, 292096516226397308078989506180479297120397, 816383155998037866924772672705065693363823, 2281716557279108884844973597531874114102906, 6377189937728384332190816053089604700753682, 17823664982455316942094008268830628363414551, 49815520081555797209727940644835038887716717, 139229840969218863387145159969110416494879097
Generating function in Maple syntax:
-(x^2+x-1)*(x-1)^3/(x^8+x^7-2*x^6-4*x^5+2*x^4+5*x^3-8*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{3}}{x^{8}+x^{7}-2 x^{6}-4 x^{5}+2 x^{4}+5 x^{3}-8 x^{2}+5 x -1}
Generating function in sympy syntax:
(x - 1)**3*(-x**2 - x + 1)/(x**8 + x**7 - 2*x**6 - 4*x**5 + 2*x**4 + 5*x**3 - 8*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^8+x^7-2*x^6-4*x^5+2*x^4+5*x^3-8*x^2+5*x-1)*F(x)+(x^2+x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+x^{7}-2 x^{6}-4 x^{5}+2 x^{4}+5 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+6)+936266101/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+6)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+5)+1540968814/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+5)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+4)-1005613107/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+4)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+3)-4527943201/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+3)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+2)-1207694171/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+2)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+1)+4249936049/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n-1)+1925704842/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n-1)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n)-3630293888/9040218877*RootOf(_Z^8+_Z^7-2*_Z^6-4*_Z^5+2*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n)
Explicit closed form in latex syntax:
\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{9040218877}+\frac{936266101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +6}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{9040218877}+\frac{1540968814 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +5}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{9040218877}-\frac{1005613107 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +4}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{9040218877}-\frac{4527943201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +3}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{9040218877}-\frac{1207694171 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +2}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{9040218877}+\frac{4249936049 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{9040218877}+\frac{1925704842 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{9040218877}-\frac{3630293888 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-4 \textit{\_Z}^{5}+2 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{9040218877}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 156
a(7) = 430
a(n+8) = a(n)+a(n+1)-2*a(n+2)-4*a(n+3)+2*a(n+4)+5*a(n+5)-8*a(n+6)+5*a(n+7), n >= 8
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) = 56
a \! \left(6\right) = 156
a \! \left(7\right) = 430
a \! \left(n +8\right) = a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)-4 a \! \left(n +3\right)+2 a \! \left(n +4\right)+5 a \! \left(n +5\right)-8 a \! \left(n +6\right)+5 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18592/
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[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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[71,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[33,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[4,x]
F[32,x] = F[13,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = 2*F[22,x]+F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[68,x]
F[45,x] = F[22,x]+F[46,x]+F[66,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[56,x]
F[48,x] = F[4,x]+F[49,x]
F[49,x] = F[22,x]+F[50,x]+F[55,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[52,x]
F[55,x] = F[14,x]*F[4,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[58,x]
F[58,x] = F[2,x]*F[4,x]
F[59,x] = 2*F[22,x]+F[60,x]+F[65,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[62,x]
F[65,x] = F[4,x]*F[45,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[2,x]+F[45,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[34,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[2,x]+F[74,x]
F[74,x] = F[22,x]+F[23,x]+F[75,x]
F[75,x] = F[4,x]*F[73,x]
F[76,x] = F[45,x]+F[77,x]
F[77,x] = 2*F[22,x]+F[69,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[89,x]
F[80,x] = F[81,x]+F[83,x]
F[81,x] = F[82,x]
F[82,x] = F[11,x]*F[4,x]
F[83,x] = F[84,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[81,x]+F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[86,x]
F[89,x] = F[90,x]+F[92,x]
F[90,x] = F[91,x]
F[91,x] = F[34,x]*F[4,x]
F[92,x] = F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[90,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[4,x]*F[95,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_{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_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \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_{11}\! \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_{71}\! \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_{33}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \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_{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_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \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_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{68}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{66}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)+F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \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_{34}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{2}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{45}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{89}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \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_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{92}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{90}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \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_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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_71(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_33(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_4(x))
Eq(F_32(x), F_13(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), 2*F_22(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_68(x))
Eq(F_45(x), F_22(x) + F_46(x) + F_66(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_56(x))
Eq(F_48(x), F_4(x) + F_49(x))
Eq(F_49(x), F_22(x) + F_50(x) + F_55(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_52(x))
Eq(F_55(x), F_14(x)*F_4(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_2(x)*F_4(x))
Eq(F_59(x), 2*F_22(x) + F_60(x) + F_65(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_62(x))
Eq(F_65(x), F_4(x)*F_45(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_2(x) + F_45(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_34(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_2(x) + F_74(x))
Eq(F_74(x), F_22(x) + F_23(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_73(x))
Eq(F_76(x), F_45(x) + F_77(x))
Eq(F_77(x), 2*F_22(x) + F_69(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_89(x))
Eq(F_80(x), F_81(x) + F_83(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_11(x)*F_4(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_81(x) + F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_86(x))
Eq(F_89(x), F_90(x) + F_92(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_34(x)*F_4(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_90(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_4(x)*F_95(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": 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