0123_0132_0321_1032_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 143, 393, 1090, 3019, 8344, 23059, 63745, 176239, 487237, 1346988, 3723818, 10294755, 28460589, 78681268, 217519726, 601348154, 1662468268, 4596007693, 12705978620, 35126549601, 97109756520, 268466585254, 742194296132, 2051847057622, 5672471978949, 15681938005343, 43353793640462, 119854537262216, 331346092141695, 916029007207171, 2532425044225219, 7001062798407575, 19354918487718838, 53508000206972489, 147926538051097688, 408953064497665365, 1130578807328417017, 3125562688105102897, 8640832513356135761, 23888174378335117370, 66040497168264823194, 182573485824309243517, 504736928934828835725, 1395379872824018922714, 3857623403128842676462, 10664664590761323581428, 29483197022599485514272, 81508321173686809759059, 225335346619988959008852, 622955027231532572933951, 1722201917160804926178408, 4761145369760362742305920, 13162513063138080526507776, 36388670515641217787633704, 100598976467812051826719969, 278112773095718913769729681, 768861844073919770671660706, 2125571323792780780066047366, 5876287771793524804184347379, 16245400749627570680432474837, 44911525058863837980227215711, 124160992652594037088633286529, 343251583558394083951953377858, 948942555130929872100196062427, 2623416805840389833275452459072, 7252615766837656040933136403363, 20050353929379532193242239307177, 55430579203105337529217346671591, 153241639604653874543906114183525, 423647027440894138767538427534228, 1171201276118785977873694482822690, 3237866290407642785543601756649211, 8951303527690880371557770797723733, 24746492800591673069096910711505084, 68413377340552564097238923965065462, 189133475876990772754557186570005906, 522872471552554549003467437989426908, 1445516824770295416608659756566252933, 3996230446957039420376936258134329380, 11047853274017894319602866008103156449, 30542548430150637980599455253924508544, 84436970827803035718832686202495969542, 233431799539593860150180421514189760468, 645338226870056907905881934556486654398, 1784081808396247890266259317962942886805, 4932216575001272382017694050158755361679, 13635451148165209698761582269623549462686, 37696140302588389798403877084400047235976, 104213566406538645469777887524476171434799, 288105554998274202844137014959762252047203, 796487575303397846570543893984446886584963, 2201944553330413795353390591573468810186511, 6087426805238692316164327565519066322499302, 16829109094999987596733226456748187259781753, 46525226831093879148622658669697150216041136
Generating function in Maple syntax:
(x^4+3*x^3+x^2-1)/(x^9-x^8+5*x^6+7*x^5+5*x^4+5*x^3+2*x^2+x-1)
Generating function in latex syntax:
\frac{x^{4}+3 x^{3}+x^{2}-1}{x^{9}-x^{8}+5 x^{6}+7 x^{5}+5 x^{4}+5 x^{3}+2 x^{2}+x -1}
Generating function in sympy syntax:
(x**4 + 3*x**3 + x**2 - 1)/(x**9 - x**8 + 5*x**6 + 7*x**5 + 5*x**4 + 5*x**3 + 2*x**2 + x - 1)
Implicit equation for the generating function in Maple syntax:
(x^9-x^8+5*x^6+7*x^5+5*x^4+5*x^3+2*x^2+x-1)*F(x)-x^4-3*x^3-x^2+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{9}-x^{8}+5 x^{6}+7 x^{5}+5 x^{4}+5 x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)-x^{4}-3 x^{3}-x^{2}+1 = 0
Explicit closed form in Maple syntax:
-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+7)-14387596885/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+7)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+6)-189835102969/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+6)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+5)+215791351916/5999897379979*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+5)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+4)-201457826791/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+4)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+3)-1265734070237/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+3)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+2)-446926415475/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+2)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n+1)+1232949747723/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n+1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n-1)+1285483749123/23999589519916*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n-1)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 1)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 2)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 3)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 4)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 5)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 6)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 7)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 8)^(-n)+1703915866009/11999794759958*RootOf(_Z^9-_Z^8+5*_Z^6+7*_Z^5+5*_Z^4+5*_Z^3+2*_Z^2+_Z-1,index = 9)^(-n)
Explicit closed form in latex syntax:
-\frac{14387596885 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf}\! 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\left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =9\right)^{-n -1}}{23999589519916}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf}\! \left(\textit{\_Z}^{9}-\textit{\_Z}^{8}+5 \textit{\_Z}^{6}+7 \textit{\_Z}^{5}+5 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =9\right)^{-n}}{11999794759958}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 143
a(7) = 393
a(8) = 1090
a(n+1) = a(n)+5*a(n+3)+7*a(n+4)+5*a(n+5)+5*a(n+6)+2*a(n+7)+a(n+8)-a(n+9), n >= 9
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 143
a \! \left(7\right) = 393
a \! \left(8\right) = 1090
a \! \left(n +1\right) = a \! \left(n \right)+5 a \! \left(n +3\right)+7 a \! \left(n +4\right)+5 a \! \left(n +5\right)+5 a \! \left(n +6\right)+2 a \! \left(n +7\right)+a \! \left(n +8\right)-a \! \left(n +9\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16225/
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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[55,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[18,x]
F[28,x] = F[11,x]*F[4,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[2,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[51,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[33,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[15,x]+F[21,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[2,x]+F[44,x]
F[44,x] = F[15,x]+F[34,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[4,x]+F[48,x]
F[48,x] = F[15,x]+F[18,x]+F[25,x]
F[49,x] = F[33,x]+F[50,x]
F[50,x] = 2*F[15,x]+F[36,x]+F[40,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[41,x]+F[53,x]
F[53,x] = 2*F[15,x]+F[40,x]+F[54,x]
F[54,x] = F[38,x]*F[4,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[43,x]+F[57,x]
F[57,x] = F[41,x]+F[58,x]
F[58,x] = F[15,x]+F[40,x]+F[54,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[65,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[16,x]
F[63,x] = F[64,x]
F[64,x] = F[27,x]*F[4,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[33,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{55}\! \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_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \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_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{51}\! \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_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \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_{2}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{49}\! \left(x \right) = F_{33}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{41}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \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_{65}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{33}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_55(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_18(x))
Eq(F_28(x), F_11(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_2(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_51(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_33(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_15(x) + F_21(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_2(x) + F_44(x))
Eq(F_44(x), F_15(x) + F_34(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_4(x) + F_48(x))
Eq(F_48(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_49(x), F_33(x) + F_50(x))
Eq(F_50(x), 2*F_15(x) + F_36(x) + F_40(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_41(x) + F_53(x))
Eq(F_53(x), 2*F_15(x) + F_40(x) + F_54(x))
Eq(F_54(x), F_38(x)*F_4(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_43(x) + F_57(x))
Eq(F_57(x), F_41(x) + F_58(x))
Eq(F_58(x), F_15(x) + F_40(x) + F_54(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_65(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_16(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_27(x)*F_4(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_33(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_40(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:
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