0123_0132_1302_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 51, 136, 373, 1031, 2835, 7778, 21356, 58673, 161185, 442741, 1216110, 3340480, 9175870, 25204798, 69233826, 190175179, 522383652, 1434912018, 3941494364, 10826711103, 29739399820, 81689803712, 224390002223, 616366678918, 1693069564145, 4650615694371, 12774564490259, 35089869518234, 96386764790279, 264760415315288, 727258329196688, 1997672789389458, 5487316422858982, 15072859621726812, 41402951765265788, 113727882956309791, 312393943191589733, 858100697964127543, 2357077734362319757, 6474549501017729056, 17784645211316965372, 48851832122482228394, 134188873231194875183, 368597305704135635053, 1012483155278210566560, 2781143876686288929739, 7639397477881277918652, 20984313078615257494968, 57640854093046654191643, 158331037481601514745673, 434912317390945798948859, 1194640841283834559387631, 3281504529981976998678663, 9013815372928308803709095, 24759639011586234108740207, 68011124992113674932402812, 186816662412905443981724771, 513158183446376014934704910, 1409570847893455145569013095, 3871885978485423526866838173, 10635507291312244955381247005, 29214190699851923737644334951, 80247130190440661330259509582, 220427187936244048837844670789, 605481405580136942421888383291, 1663169302914372015522831202674, 4568484027195404594541705390334, 12548960752322209306357609178250, 34470168884446788065409158521451, 94684537339269804618585873733778, 260084644238477446063811781844687, 714414666528665129873428578572120, 1962393117231778212471452663858521, 5390408297845518647081494316271444, 14806667105758075707862891790590273, 40671759663987749890012126254941455, 111719404667502258879351888799334788, 306876945634404277473255863428621710, 842946308586040733256378088029095327, 2315450832221692955706219926331344290, 6360206458972699414976080722962588019, 17470561515635303089150198590147709484, 47989089920344585602442097083054562546, 131819045960365244632157378598237232535, 362087735081934941998917239885539724742, 994602312143772603757965598399358995002, 2732027802869075177741186205093984975200, 7504482771170842876666559566287457855692, 20613722013977198599097235286381668446705, 56622894372136943447824350718802818589781, 155534850276152751635806948519309790828874, 427231598078271884665729591135595177316248, 1173543023138781525373710165436885216345699, 3223551893990287742261255668291849959947059, 8854627915946045834530068959086603499476544, 24322374234465244847048571373163772004575139
Generating function in Maple syntax:
-(x^3+x^2+x-1)/(x-1)/(x^7+4*x^6+7*x^5+7*x^4+4*x^3+2*x^2+x-1)
Generating function in latex syntax:
-\frac{x^{3}+x^{2}+x -1}{\left(x -1\right) \left(x^{7}+4 x^{6}+7 x^{5}+7 x^{4}+4 x^{3}+2 x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**3 - x**2 - x + 1)/((x - 1)*(x**7 + 4*x**6 + 7*x**5 + 7*x**4 + 4*x**3 + 2*x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^7+4*x^6+7*x^5+7*x^4+4*x^3+2*x^2+x-1)*F(x)+x^3+x^2+x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{7}+4 x^{6}+7 x^{5}+7 x^{4}+4 x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+x^{3}+x^{2}+x -1 = 0
Explicit closed form in Maple syntax:
2/25+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+6)+34910219/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+6)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+5)+27197076/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+5)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+4)-59705592/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+4)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+3)-201919067/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+3)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+2)-106787574/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+2)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+1)+130289913/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n-1)+96897856/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n-1)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n)+202817719/1546256875*RootOf(_Z^7+4*_Z^6+7*_Z^5+7*_Z^4+4*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{2}{25}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{1546256875}+\frac{34910219 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{1546256875}+\frac{27197076 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{1546256875}-\frac{59705592 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{1546256875}-\frac{201919067 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{1546256875}-\frac{106787574 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{1546256875}+\frac{130289913 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{1546256875}+\frac{96897856 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{1546256875}+\frac{202817719 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+4 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+7 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{1546256875}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 136
a(n+7) = a(n)+4*a(n+1)+7*a(n+2)+7*a(n+3)+4*a(n+4)+2*a(n+5)+a(n+6)-2, 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) = 19
a \! \left(5\right) = 51
a \! \left(6\right) = 136
a \! \left(n +7\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+7 a \! \left(n +2\right)+7 a \! \left(n +3\right)+4 a \! \left(n +4\right)+2 a \! \left(n +5\right)+a \! \left(n +6\right)-2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16576/
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[19,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[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[44,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[16,x]
F[27,x] = F[28,x]+F[41,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[2,x]*F[4,x]
F[34,x] = F[28,x]+F[35,x]
F[35,x] = 2*F[13,x]+F[36,x]+F[40,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[32,x]
F[39,x] = F[40,x]
F[40,x] = F[28,x]*F[4,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[32,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[54,x]
F[46,x] = F[2,x]+F[47,x]
F[47,x] = F[13,x]+F[33,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[4,x]+F[51,x]
F[51,x] = F[25,x]
F[52,x] = F[32,x]+F[53,x]
F[53,x] = F[42,x]
F[54,x] = F[28,x]+F[55,x]
F[55,x] = 2*F[13,x]+F[36,x]+F[40,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_{19}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \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_{16}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{46}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{25}\! \left(x \right)
F_{52}\! \left(x \right) = F_{32}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{42}\! \left(x \right)
F_{54}\! \left(x \right) = F_{28}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \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_19(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) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_44(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_16(x))
Eq(F_27(x), F_28(x) + F_41(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_2(x)*F_4(x))
Eq(F_34(x), F_28(x) + F_35(x))
Eq(F_35(x), 2*F_13(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_32(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_28(x)*F_4(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_32(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_54(x))
Eq(F_46(x), F_2(x) + F_47(x))
Eq(F_47(x), F_13(x) + F_33(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_4(x) + F_51(x))
Eq(F_51(x), F_25(x))
Eq(F_52(x), F_32(x) + F_53(x))
Eq(F_53(x), F_42(x))
Eq(F_54(x), F_28(x) + F_55(x))
Eq(F_55(x), 2*F_13(x) + F_36(x) + F_40(x))
Pack JSON:
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Specification JSON:
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