0123_0132_0321_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 54, 150, 424, 1212, 3468, 9913, 28336, 81035, 231805, 663137, 1897106, 5427361, 15527219, 44422508, 127090901, 363602494, 1040255197, 2976139918, 8514654456, 24360199223, 69693891205, 199392413995, 570456553139, 1632061534958, 4669286181986, 13358708113420, 38218921808969, 109343356910161, 312828545579556, 894994463319962, 2560556257373865, 7325685932378022, 20958600005830742, 59962018343412004, 171549800227757404, 490799589012345050, 1404165066121280713, 4017280325982908876, 11493336223085932251, 32882140856106042859, 94074963639494926235, 269146063892630617280, 770020002203209944235, 2203007523936234613809, 6302748158014832580106, 18032001212772433399953, 51589094088104222528066, 147595078185012802059087, 422265742198093695908910, 1208091483989711557496042, 3456318824470968152885693, 9888439720591698799668169, 28290573026851052790567649, 80938605563922308492398881, 231563279556567605466582646, 662496617842579408143747726, 1895385872463597565726726734, 5422650484214593235605290207, 15514064286937178848877792803, 44385340969302792195868694812, 126985324833290498818446395396, 363301765196952181360301902985, 1039397054490341629080400193068, 2973688378027901404346817201484, 8507646362298192449156950577230, 24340158558889368061145126032766, 69636570849641794228135893155958, 199228447422178615389932235249211, 569987490451735272419643093509058, 1630719626013106725742073580849233, 4665447125088271039262702497526351, 13347724850905466682781008936064319, 38187499272560829321454151815332244, 109253458321993242326792462530182031, 312571348810269256010940416783336501, 894258631238251293562929716865156214, 2558451062734887885976845511546513167, 7319663027848772358544719005715163660, 20941368635749461095599778147080536299, 59912719843776352692248955432728589440, 171408758496856886613557371170370015926, 490396072253859797320520279404760288005, 1403010615040553487232297227686367640927, 4013977471046066339367580839006465587601, 11483886839729764039922525428061898417081, 32855106412281344054128986504766901900340, 93997618787728513369413958955415114427940, 268924782251211173061962069028725227365954, 769386921089780470931602642249420868758475, 2201196295070518929623019749708278419369659, 6297566278575692830134018683262005070346266, 18017175988288290675087000123858767671409296, 51546679500191056186307073786872448066003179, 147473731134256908302885679913882422950915164, 421918571390781538600284200923468234049985312, 1207098237193013991679232022622663403488850154
Generating function in Maple syntax:
(x-1)*(x^3+2*x-1)/(x^8+2*x^7+3*x^6+3*x^5+x^4-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{8}+2 x^{7}+3 x^{6}+3 x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(x - 1)*(x**3 + 2*x - 1)/(x**8 + 2*x**7 + 3*x**6 + 3*x**5 + x**4 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8+2*x^7+3*x^6+3*x^5+x^4-3*x^3+4*x^2-4*x+1)*F(x)-(x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+2 x^{7}+3 x^{6}+3 x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+6)-121435339/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+6)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+5)-16926043/207528668*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+5)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+4)-1370323/8584819*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+4)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+3)-980003181/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+3)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+2)-158406304/1193289841*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+2)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n+1)+164492019/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n+1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n-1)+332574729/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n-1)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 6)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 7)^(-n)+409636205/4773159364*RootOf(_Z^8+2*_Z^7+3*_Z^6+3*_Z^5+_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{4773159364}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{207528668}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{8584819}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{4773159364}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{1193289841}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+2 \textit{\_Z}^{7}+3 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{4773159364}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 150
a(7) = 424
a(n+8) = -a(n)-2*a(n+1)-3*a(n+2)-3*a(n+3)-a(n+4)+3*a(n+5)-4*a(n+6)+4*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) = 54
a \! \left(6\right) = 150
a \! \left(7\right) = 424
a \! \left(n +8\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right)-4 a \! \left(n +6\right)+4 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16260/
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[52,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[28,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[17,x]+F[27,x]
F[27,x] = F[16,x]
F[28,x] = F[20,x]+F[29,x]
F[29,x] = 2*F[15,x]+F[30,x]+F[34,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[24,x]
F[33,x] = F[29,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[49,x]
F[36,x] = F[37,x]
F[37,x] = F[15,x]+F[38,x]+F[48,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[4,x]+F[41,x]
F[41,x] = F[25,x]
F[42,x] = F[37,x]+F[43,x]
F[43,x] = 2*F[15,x]+F[34,x]+F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[41,x]
F[47,x] = F[43,x]
F[48,x] = F[2,x]*F[4,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[37,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[2,x]+F[37,x]
F[55,x] = F[56,x]+F[60,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[2,x]+F[59,x]
F[59,x] = F[48,x]
F[60,x] = 2*F[15,x]+F[50,x]+F[61,x]
F[61,x] = F[4,x]*F[56,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_{52}\! \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_{28}\! \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_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{24}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{25}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \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_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{48}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \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_52(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_28(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_17(x) + F_27(x))
Eq(F_27(x), F_16(x))
Eq(F_28(x), F_20(x) + F_29(x))
Eq(F_29(x), 2*F_15(x) + F_30(x) + F_34(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_24(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_49(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_15(x) + F_38(x) + F_48(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_4(x) + F_41(x))
Eq(F_41(x), F_25(x))
Eq(F_42(x), F_37(x) + F_43(x))
Eq(F_43(x), 2*F_15(x) + F_34(x) + F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_2(x)*F_4(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_37(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_2(x) + F_37(x))
Eq(F_55(x), F_56(x) + F_60(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_2(x) + F_59(x))
Eq(F_59(x), F_48(x))
Eq(F_60(x), 2*F_15(x) + F_50(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_56(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": 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