0123_0231_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 60, 166, 456, 1272, 3584, 10104, 28416, 79812, 224172, 629868, 1770068, 4974204, 13977668, 39276868, 110367364, 310132804, 871476804, 2448858116, 6881307460, 19336513684, 54335722516, 152683736836, 429042316420, 1205611736308, 3387776856820, 9519675072644, 26750349166884, 75168656175572, 211224415712916, 593541989946852, 1667856874017860, 4686688725108340, 13169625973368820, 37006735129881348, 103989167782075684, 292210241674120084, 821112690484222932, 2307332031252175780, 6483618100351699204, 18219009272118024756, 51195535227386363316, 143859788865082389060, 404246947714953031396, 1135936567306645777364, 3191989184428721188500, 8969510487428100119780, 25204383140314747031748, 70824481489171271785332, 199017256255981453554036, 559239791875001258164100, 1571467473223168195268772, 4415833878913936090178836, 12408522084246098030312020, 34868028222359673918250020, 97979387380776665909155460, 275322719434882518737757364, 773658642530881303264666932, 2173985846105557095053182660, 6108914447858240946778469988, 17166089557622241410606273108, 48236823942364917911292996116, 135545790800884330863703660388, 380884558771726183865277874244, 1070288101560037884025400694260, 3007516566266282408349163464948, 8451141223734085609530513600004, 23747762118619048947207022154788, 66731366890273934316718154538900, 187515577459527716079654740283860, 526919999223108936342597058620068, 1480648644463718511084753669984772, 4160632376043052489175721956781876, 11691404191875338652133464175142068, 32852922254043874308940433449069892, 92316926428759943390440934916176356, 259411167120880699163420988396815572, 728947076449168035092023627930214804, 2048346052952237498774452239726032356, 5755865807272487930746341158235950020, 16174020568242860696494425984280273012, 45449103592966857059364853357771688052, 127712278384263309963975592515119973508, 358872337641078798534691284246147647396, 1008433616198345655021970582585046385172, 2833705057802335080525114208048450738516, 7962729748028513115525824071631312918820, 22375322677128469283909249290368495988100, 62874803082394750284343053322366618511796, 176678607933141746363838060156784091911220, 496468043968049603687351060723362087400388, 1395078451007117643784398366728804547362148, 3920179572705126686654875090843111632754516, 11015730241665200522837822042497965961492756, 30954274034289205878588340696074251801400420, 86981712511964068942193559654968398077081412, 244419181116411486125846547490047264421825268
Generating function in Maple syntax:
(x-1)^3/(2*x^7-2*x^6+2*x^4+4*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{2 x^{7}-2 x^{6}+2 x^{4}+4 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**3/(2*x**7 - 2*x**6 + 2*x**4 + 4*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^7-2*x^6+2*x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{7}-2 x^{6}+2 x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+5)+716124/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+5)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+4)+757328/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+4)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+3)+240734/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+3)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+2)-107420/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+2)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+1)+1378505/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n-1)+921663/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n-1)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n)+3616444/20386583*RootOf(2*_Z^7-2*_Z^6+2*_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{20386583}+\frac{716124 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{20386583}+\frac{757328 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{20386583}+\frac{240734 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{20386583}-\frac{107420 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{20386583}+\frac{1378505 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{20386583}+\frac{921663 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{20386583}+\frac{3616444 \mathit{RootOf}\! \left(2 \textit{\_Z}^{7}-2 \textit{\_Z}^{6}+2 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{20386583}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 166
a(n+1) = a(n)+a(n+3)+2*a(n+4)-5/2*a(n+5)+2*a(n+6)-1/2*a(n+7), 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) = 60
a \! \left(6\right) = 166
a \! \left(n +1\right) = a \! \left(n \right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)-\frac{5 a \! \left(n +5\right)}{2}+2 a \! \left(n +6\right)-\frac{a \! \left(n +7\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15015/
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[17,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[63,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[30,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]+F[27,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[13,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[7,x]
F[36,x] = F[33,x]
F[37,x] = F[38,x]+F[48,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[2,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[41,x]
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[42,x]
F[48,x] = 2*F[19,x]+F[49,x]+F[56,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[42,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[42,x]+F[55,x]
F[55,x] = F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[44,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[62,x]
F[61,x] = F[38,x]
F[62,x] = F[59,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[79,x]
F[65,x] = F[2,x]+F[66,x]
F[66,x] = F[19,x]+F[43,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[74,x]
F[69,x] = F[11,x]+F[70,x]
F[70,x] = F[19,x]+F[24,x]+F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[7,x]+F[73,x]
F[73,x] = F[71,x]
F[74,x] = F[42,x]+F[75,x]
F[75,x] = 2*F[19,x]+F[49,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[38,x]+F[78,x]
F[78,x] = F[76,x]
F[79,x] = F[38,x]+F[80,x]
F[80,x] = F[46,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_{17}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{42}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{44}\! \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_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{38}\! \left(x \right)
F_{62}\! \left(x \right) = F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{79}\! \left(x \right)
F_{65}\! \left(x \right) = F_{2}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \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_{74}\! \left(x \right)
F_{69}\! \left(x \right) = F_{11}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{7}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{42}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{38}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{38}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{46}\! \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_17(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_63(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_30(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x) + F_27(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_13(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_7(x))
Eq(F_36(x), F_33(x))
Eq(F_37(x), F_38(x) + F_48(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_2(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_41(x))
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_42(x))
Eq(F_48(x), 2*F_19(x) + F_49(x) + F_56(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_42(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_42(x) + F_55(x))
Eq(F_55(x), F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_44(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) + F_62(x))
Eq(F_61(x), F_38(x))
Eq(F_62(x), F_59(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_79(x))
Eq(F_65(x), F_2(x) + F_66(x))
Eq(F_66(x), F_19(x) + F_43(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_74(x))
Eq(F_69(x), F_11(x) + F_70(x))
Eq(F_70(x), F_19(x) + F_24(x) + F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_7(x) + F_73(x))
Eq(F_73(x), F_71(x))
Eq(F_74(x), F_42(x) + F_75(x))
Eq(F_75(x), 2*F_19(x) + F_49(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_38(x) + F_78(x))
Eq(F_78(x), F_76(x))
Eq(F_79(x), F_38(x) + F_80(x))
Eq(F_80(x), F_46(x))
Pack JSON:
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Specification JSON:
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