0132_0231_0321_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 142, 385, 1057, 2906, 7971, 21842, 59862, 164108, 449913, 1233411, 3381245, 9269289, 25410825, 69661320, 190969612, 523523992, 1435188404, 3934425243, 10785833077, 29568281567, 81058482688, 222213712234, 609176638798, 1669996751651, 4578128858178, 12550481799119, 34405889013708, 94320299238770, 258569654896857, 708842815097229, 1943221592310650, 5327147396281788, 14603841113146272, 40034967946821840, 109751855425224628, 300873720825198214, 824815175401497863, 2261148204325458609, 6198711365168009404, 16993146453275327825, 46585009265814773722, 127708137763843052035, 350099070669863260183, 959761542452026828895, 2631090155730556733022, 7212870178040363759675, 19773361278385561625526, 54206689791245194413578, 148602211670316232152795, 407378081900068092843942, 1116786215677360670467973, 3061557572537512292163138, 8392953493141682659429995, 23008441510265302587639618, 63075338277984045131176873, 172914723368251488501227821, 474028398004725805954385576, 1299501382750290256918460379, 3562452905517869237336433439, 9766107887606150285040997679, 26772806771602290109167318268, 73394968669064126496627092350, 201204956652010977929440414171, 551583239498013792332770943048, 1512110214169927221142405611402, 4145298725679021640382700423464, 11363921335951693486684846856670, 31153052331239294549141663966201, 85402973222152860971549642598228, 234123698622940373710585206244158, 641826674046836523524117828503538, 1759503552784128394880217917426595, 4823502788907233847381605946611256, 13223149858254571830290696059786146, 36249941137376352962888459920252753, 99375583469089061737387914363417658, 272428210363064955452256944400652622, 746834657073563064554708629490504067, 2047372422491992895333218592014571329, 5612666467308635504355968735597240447, 15386563053783635791635114457795145815, 42180721763355252570586191378848550850, 115634224632126253613474248077738341131, 316999646930873667738454857636271457732, 869022786929987479721885335063330011009, 2382338944270952993938923482433662789360, 6530943642387235467485460352983042151722, 17903927970707398764615954212752198899506, 49081825587934315584378594497215209547848, 134552909673553672233785875570144535386782, 368863327407896618418592378926907598787545, 1011201873200128297386815007713487497143560, 2772108671114151162333977168951065533142647, 7599458316020542458970762208361687860046639, 20833153944763102052082548979640178480430073
Generating function in Maple syntax:
(x-1)*(x^2+x-1)/(x^7+3*x^6+x^5-3*x^4+x^2-3*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{2}+x -1\right)}{x^{7}+3 x^{6}+x^{5}-3 x^{4}+x^{2}-3 x +1}
Generating function in sympy syntax:
(x - 1)*(x**2 + x - 1)/(x**7 + 3*x**6 + x**5 - 3*x**4 + x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7+3*x^6+x^5-3*x^4+x^2-3*x+1)*F(x)-(x-1)*(x^2+x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+3 x^{6}+x^{5}-3 x^{4}+x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}+x -1\right) = 0
Explicit closed form in Maple syntax:
-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n+5)-19874/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n+5)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n+4)-822233/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n+4)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n+3)-1762309/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n+3)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n+2)-493117/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n+2)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n+1)+723336/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n+1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n-1)+867807/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n-1)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 1)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 2)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 3)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 4)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 5)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 6)^(-n)+876467/10092073*RootOf(_Z^7+3*_Z^6+_Z^5-3*_Z^4+_Z^2-3*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{10092073}-\frac{19874 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{10092073}-\frac{822233 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{10092073}-\frac{1762309 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{10092073}-\frac{493117 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{10092073}+\frac{723336 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{10092073}+\frac{867807 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{10092073}+\frac{876467 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+3 \textit{\_Z}^{6}+\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{10092073}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 142
a(n+3) = 1/3*a(n)+a(n+1)+1/3*a(n+2)+1/3*a(n+5)-a(n+6)+1/3*a(n+7), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 142
a \! \left(n +3\right) = \frac{a \! \left(n \right)}{3}+a \! \left(n +1\right)+\frac{a \! \left(n +2\right)}{3}+\frac{a \! \left(n +5\right)}{3}-a \! \left(n +6\right)+\frac{a \! \left(n +7\right)}{3}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18701/
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[17,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[53,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[27,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]
F[27,x] = F[28,x]+F[38,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[31,x]*F[4,x]
F[34,x] = F[32,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[32,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[19,x]+F[43,x]+F[51,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[11,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[26,x]*F[4,x]
F[48,x] = F[32,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[41,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[2,x]+F[42,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[52,x]+F[55,x]
F[55,x] = F[42,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[32,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_{17}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{42}\! \left(x \right)+F_{56}\! \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_{32}\! \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_17(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_53(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_27(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x))
Eq(F_27(x), F_28(x) + F_38(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_31(x)*F_4(x))
Eq(F_34(x), F_32(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_32(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_19(x) + F_43(x) + F_51(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_11(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_26(x)*F_4(x))
Eq(F_48(x), F_32(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_41(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_2(x) + F_42(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_52(x) + F_55(x))
Eq(F_55(x), F_42(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_32(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": [0, 3, 2, 1], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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": [0, 3, 2, 1], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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": [0, 3, 2, 1], "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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "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, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 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