0132_0231_1023_1302_2013
Counting sequence:
1, 1, 2, 6, 19, 57, 167, 488, 1426, 4163, 12144, 35418, 103302, 301320, 878954, 2563963, 7479279, 21817662, 63643864, 185654123, 541567329, 1579793095, 4608376214, 13442982480, 39214198273, 114390787898, 333686598316, 973389101216, 2839449795920, 8282890301600, 24161818901169, 70481857335957, 205600920755282, 599753471483994, 1749526340787355, 5103500959354045, 14887299170584399, 43427380216014384, 126680960113578014, 369537963733470507, 1077970253129386908, 3144520944186452534, 9172805965398477894, 26757770347946579352, 78054444484517122578, 227690731498264646915, 664191123933990705571, 1937495857691772727614, 5651822289250962942932, 16486794055564725345355, 48093228045680112955693, 140291591928578021711455, 409241208495309858435294, 1193787627814248126716372, 3482369005707553557170921, 10158334371513142212027078, 29632631416812944009508840, 86440632151970848682169632, 252153876641305888937866512, 735551972751153069385128960, 2145660863218648600963233537, 6259055390373812488123741897, 18258139042999062093594312722, 53260375651281353089351248574, 155364553191049402486707544595, 453210179107583834887149666897, 1322048448169201459125148772519, 3856515541526914879025281571624, 11249748178771100311542861916234, 32816368227484411332653756047715, 95727833772674569321824280253496, 279245347787569005471948010043682, 814579847760632344121886356639782, 2376191165349374482752759495190040, 6931529757097062844494635220624826, 20219797747820422377223410752597131, 58982682797280069169381369256708983, 172056956916869739624660976061285390, 501903186147692686628734602709503648, 1464089640890926654117423854842780347, 4270860468164766167912725569463923465, 12458423739295787840151567720205438887, 36342166461491825332675504421584227830, 106012854495302051909235822430836185336, 309247532894073403338701771602521816961, 902098496039544181174610565842078679778, 2631489696752252689608802826715305296308, 7676254926169072788267523352316583598080, 22392217520083480223447811314311048725376, 65319795953802298854653112774864016203232, 190542796380912461676994649002059587241009, 555827781188046466450320134449698168536093, 1621391772391219293407143547944572594121154, 4729722710079026848765683373620049353704946, 13796959683128117813231133190310590944545659, 40246777277706869440435561725564040399290613, 117402900236212900568675128350527598468033071, 342473159745611681745764082713653274224509648, 999020168242537500848020719019345213994211094, 2914217561740285978334405299178749728349823787, 8500993540596557596157504193430310198521241092
Generating function in Maple syntax:
(2*x-1)*(x-1)^2/(x^8+2*x^7+x^6-x^5-x^4+6*x^3-8*x^2+5*x-1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{8}+2 x^{7}+x^{6}-x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1}
Generating function in sympy syntax:
(x - 1)**2*(2*x - 1)/(x**8 + 2*x**7 + x**6 - x**5 - x**4 + 6*x**3 - 8*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^8+2*x^7+x^6-x^5-x^4+6*x^3-8*x^2+5*x-1)*F(x)-(2*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+2 x^{7}+x^{6}-x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+6)+352012550/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+6)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+5)+775052961/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+5)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+4)+380434700/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+4)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+3)-652934211/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+3)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+2)-865759159/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+2)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n+1)+1986763722/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n+1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n-1)+930863637/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n-1)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 6)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 7)^(-n)-1491669994/6565172363*RootOf(_Z^8+2*_Z^7+_Z^6-_Z^5-_Z^4+6*_Z^3-8*_Z^2+5*_Z-1,index = 8)^(-n)
Explicit closed form in latex syntax:
\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +6}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +5}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +4}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +3}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +2}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+\textit{\_Z}^{6}-\textit{\_Z}^{5}-\textit{\_Z}^{4}+6 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{6565172363}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 167
a(7) = 488
a(n+8) = a(n)+2*a(n+1)+a(n+2)-a(n+3)-a(n+4)+6*a(n+5)-8*a(n+6)+5*a(n+7), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 167
a \! \left(7\right) = 488
a \! \left(n +8\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)-a \! \left(n +3\right)-a \! \left(n +4\right)+6 a \! \left(n +5\right)-8 a \! \left(n +6\right)+5 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18779/
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[12,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[12,x]+F[21,x]
F[21,x] = F[18,x]+F[19,x]+F[22,x]
F[22,x] = F[12,x]*F[14,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[14,x]+F[25,x]
F[25,x] = F[23,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[18,x]+F[28,x]+F[57,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[41,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[12,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[12,x]+F[38,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[18,x]+F[28,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[2,x]+F[42,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[53,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[2,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[48,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[49,x]+F[54,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[2,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[59,x]
F[62,x] = F[42,x]+F[63,x]
F[63,x] = 2*F[18,x]+F[64,x]+F[68,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[49,x]+F[66,x]
F[66,x] = 2*F[18,x]+F[64,x]+F[67,x]
F[67,x] = F[12,x]*F[42,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[42,x]+F[70,x]
F[70,x] = F[68,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{57}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{42}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{49}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{42}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{68}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_26(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_12(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_19(x) + F_22(x))
Eq(F_22(x), F_12(x)*F_14(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_14(x) + F_25(x))
Eq(F_25(x), F_23(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_28(x) + F_57(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_41(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_12(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_12(x) + F_38(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_18(x) + F_28(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_2(x) + F_42(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_53(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_2(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_48(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_49(x) + F_54(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_2(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_59(x))
Eq(F_62(x), F_42(x) + F_63(x))
Eq(F_63(x), 2*F_18(x) + F_64(x) + F_68(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_49(x) + F_66(x))
Eq(F_66(x), 2*F_18(x) + F_64(x) + F_67(x))
Eq(F_67(x), F_12(x)*F_42(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_42(x) + F_70(x))
Eq(F_70(x), F_68(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 0, 1, 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, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 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], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 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