0123_0132_0231_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 130, 342, 916, 2449, 6510, 17290, 45977, 122327, 325410, 865482, 2301883, 6122500, 16284717, 43313963, 115205479, 306421091, 815013680, 2167760272, 5765771795, 15335699003, 40789626163, 108491546022, 288563948202, 767517415827, 2041429585881, 5429759219608, 14441979991185, 38412529466659, 102168983768116, 271747301945071, 722788789697936, 1922461164465409, 5113329068599151, 13600344520159048, 36173961930764298, 96214880427050822, 255910680542000164, 680666817073825420, 1810425867665746002, 4815339517216390459, 12807757047774470232, 34065851433388865782, 90607764462813187404, 240996970147703495153, 640999587228643021124, 1704919653452365460654, 4534715907221486120902, 12061359206908865600142, 32080595321619957934673, 85327414475810398725097, 226952386267596412168935, 603644045105318317231268, 1605562026395567048668385, 4270446203364241449701973, 11358459203702562053223429, 30211034008703406149946990, 80354787520609768086749596, 213726278803373519158389437, 568465472445203184436488244, 1511994665193447040443504147, 4021577348822735480628905328, 10696522114047863589617953798, 28450425147189734636707305686, 75671950417679854720744617643, 201270949393226312370855413780, 535337001967719691001544152249, 1423880130439883717960107537127, 3787211828080870522886050080599, 10073160741645176119558037486634, 26792419313508459405698311838213, 71262015079650415836692810263741, 189541479393460742176513802698154, 504139159838615417137052491439173, 1340900647689856107528198958237505, 3566504430147133265856595533834290, 9486127008867842790385175666953170, 25231037109537430474751830486947615, 67109077606460696168560535281171388, 178495567884824497061232113659044860, 474759434801987412778419056119417436, 1262756961444225915449990481753762296, 3358659200403065297789941976472633020, 8933303849341245374407228978671349352, 23760647598624480725637806416020113340, 63198160929861507324990302645562082249, 168093379119342234722826654089450938995, 447091872422004020292802339803882559623, 1189167255920863136707782184747701797161, 3162926570088905294164043719848757088674, 8412697572998193310505410032430253364686, 22375947998293350144186175170973090963736, 59515160800424473854185026592891705344212, 158297398857493775562512956802313584979892, 421036692971007693437340567596568439770037, 1119866138720008245682213632441344586422624
Generating function in Maple syntax:
(x-1)^2/(x^8-2*x^7+2*x^6+x^5-3*x^4-2*x^3+2*x^2-3*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{2}}{x^{8}-2 x^{7}+2 x^{6}+x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1}
Generating function in sympy syntax:
(x - 1)**2/(x**8 - 2*x**7 + 2*x**6 + x**5 - 3*x**4 - 2*x**3 + 2*x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8-2*x^7+2*x^6+x^5-3*x^4-2*x^3+2*x^2-3*x+1)*F(x)-(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}-2 x^{7}+2 x^{6}+x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+6)-462609344/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+6)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+5)+27816419/1689106373*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+5)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+4)-49356472/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+4)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+3)-830044855/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+3)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+2)+64931413/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+2)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+1)+565596724/5067319119*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n-1)+870658318/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n-1)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n)+2659260971/15201957357*RootOf(_Z^8-2*_Z^7+2*_Z^6+_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{15201957357}-\frac{462609344 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{15201957357}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{1689106373}+\frac{27816419 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{1689106373}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{15201957357}-\frac{49356472 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{15201957357}-\frac{830044855 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{15201957357}+\frac{64931413 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{15201957357}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{5067319119}+\frac{565596724 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{5067319119}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{15201957357}+\frac{870658318 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{15201957357}+\frac{2659260971 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-2 \textit{\_Z}^{7}+2 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{15201957357}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 130
a(7) = 342
a(n+8) = -a(n)+2*a(n+1)-2*a(n+2)-a(n+3)+3*a(n+4)+2*a(n+5)-2*a(n+6)+3*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) = 51
a \! \left(6\right) = 130
a \! \left(7\right) = 342
a \! \left(n +8\right) = -a \! \left(n \right)+2 a \! \left(n +1\right)-2 a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)+2 a \! \left(n +5\right)-2 a \! \left(n +6\right)+3 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16126/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[4,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[18,x]+F[50,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[29,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[13,x]+F[17,x]+F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[7,x]
F[28,x] = F[25,x]
F[29,x] = F[30,x]+F[40,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[2,x]*F[4,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[34,x]
F[40,x] = 2*F[17,x]+F[41,x]+F[43,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[34,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[36,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[30,x]
F[49,x] = F[46,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[62,x]
F[52,x] = F[2,x]+F[53,x]
F[53,x] = F[17,x]+F[35,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[4,x]+F[57,x]
F[57,x] = F[13,x]+F[17,x]+F[58,x]
F[58,x] = F[4,x]*F[7,x]
F[59,x] = F[34,x]+F[60,x]
F[60,x] = 2*F[17,x]+F[41,x]+F[61,x]
F[61,x] = F[30,x]*F[4,x]
F[62,x] = F[30,x]+F[63,x]
F[63,x] = F[38,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \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_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \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_{34}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{34}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \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_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{30}\! \left(x \right)
F_{49}\! \left(x \right) = F_{46}\! \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_{62}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{41}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{30}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{38}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_4(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_50(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_29(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_13(x) + F_17(x) + F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_30(x) + F_40(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_2(x)*F_4(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_34(x))
Eq(F_40(x), 2*F_17(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_34(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_36(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_48(x) + F_49(x))
Eq(F_48(x), F_30(x))
Eq(F_49(x), F_46(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_62(x))
Eq(F_52(x), F_2(x) + F_53(x))
Eq(F_53(x), F_17(x) + F_35(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_4(x) + F_57(x))
Eq(F_57(x), F_13(x) + F_17(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_7(x))
Eq(F_59(x), F_34(x) + F_60(x))
Eq(F_60(x), 2*F_17(x) + F_41(x) + F_61(x))
Eq(F_61(x), F_30(x)*F_4(x))
Eq(F_62(x), F_30(x) + F_63(x))
Eq(F_63(x), F_38(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"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], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 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