0123_0231_1032_1302_2103

Counting sequence:
1, 1, 2, 6, 19, 50, 128, 337, 900, 2395, 6347, 16819, 44610, 118349, 313925, 832623, 2208404, 5857595, 15536772, 41209744, 109304583, 289919358, 768982220, 2039648579, 5409963465, 14349385762, 38060308430, 100951156832, 267762833172, 710214095930, 1883771755235, 4996515908711, 13252758015672, 35151613284178, 93236133560859, 247299506037568, 655937171051694, 1739807649682159, 4614665537285922, 12239938147694783, 32465210024225275, 86110717979077873, 228400054869142792, 605808269731249811, 1606845759669536319, 4261997441060781751, 11304521344565453728, 29984110642245864661, 79529850367229211962, 210944962647060151554, 559510385857650664761, 1484045259740835088134, 3936281414299911389984, 10440592206243592025921, 27692625131189135576171, 73451914556912232738820, 194823846656543506049236, 516750740331525867633412, 1370629582650278463379448, 3635457690163788170335960, 9642687407501609052821813, 25576262568084243878904429, 67838474826290596623775750, 179934759994996959076721418, 477258929204430404919799307, 1265881509007446755980322282, 3357623916053690188075285972, 8905761148604785366969910573, 23621639474505749168151842040, 62654032839286742957512516471, 166183546881373184684891845499, 440785213697473975646556425299, 1169138631714359736606486845790, 3101023123486772601864460206793, 8225153246624643446992064907553, 21816395181984681131831315517019, 57865803160791213957554445510740, 153483247232727251277092621571115, 407098940900285174108040543614776, 1079789167027705089947515025880020, 2864032617358176116654907912411923, 7596559665319425715047071351097906, 20149113665468096149181203918541476, 53443505927743140705211471998656691, 141753546745023135454755382722619037, 375987085165428800086867958158140214, 997268085753619267694804996562189810, 2645153714322670174664169239860946100, 7016005297219173963328869010383968820, 18609251350525808875833105786151020018, 49359175364976694297636627158857418831, 130920269000594865051253849562220575123, 347252900974315411515444887068114463180, 921053540109437162207549732773706248966, 2443002265403317909742480157456899943531, 6479818825794437938190383775098986169448, 17187070437770615458857621674786240409706, 45586982934923737868690620734632144996635, 120914906390445556140303344633748576998342, 320714678755580822142794417829442974330507, 850663563656556922136042008009962399664115

Generating function in Maple syntax:
(x-1)^2/(x^8-x^7+2*x^5-3*x^4-2*x^3+2*x^2-3*x+1)

Generating function in latex syntax:
\frac{\left(x -1\right)^{2}}{x^{8}-x^{7}+2 x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1}

Generating function in sympy syntax:
(x - 1)**2/(x**8 - x**7 + 2*x**5 - 3*x**4 - 2*x**3 + 2*x**2 - 3*x + 1)

Implicit equation for the generating function in Maple syntax:
(x^8-x^7+2*x^5-3*x^4-2*x^3+2*x^2-3*x+1)*F(x)-(x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{8}-x^{7}+2 x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0

Explicit closed form in Maple syntax:
-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+6)-3651851/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+6)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+5)-9989451/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+5)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+4)+3984197/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+4)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+3)-906648/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+3)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+2)-2113990/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+2)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n+1)+15459818/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n+1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n-1)+4603213/123174239*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n-1)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 6)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 7)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 8)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 1)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 2)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 3)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 4)^(-n)+55216331/246348478*RootOf(_Z^8-_Z^7+2*_Z^5-3*_Z^4-2*_Z^3+2*_Z^2-3*_Z+1,index = 5)^(-n)

Explicit closed form in latex syntax:
-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{123174239}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{246348478}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{123174239}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-\textit{\_Z}^{7}+2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}-2 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{246348478}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 128
a(7) = 337
a(n+1) = a(n)+2*a(n+3)-3*a(n+4)-2*a(n+5)+2*a(n+6)-3*a(n+7)+a(n+8), 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) = 50
a \! \left(6\right) = 128
a \! \left(7\right) = 337
a \! \left(n +1\right) = a \! \left(n \right)+2 a \! \left(n +3\right)-3 a \! \left(n +4\right)-2 a \! \left(n +5\right)+2 a \! \left(n +6\right)-3 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17230/
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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[60,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[35,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[1,x]+F[4,x]
F[26,x] = F[19,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[11,x]*F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[15,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[2,x]*F[4,x]
F[41,x] = 2*F[19,x]+F[42,x]+F[50,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[2,x]+F[47,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[59,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[49,x]+F[55,x]
F[55,x] = F[52,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[39,x]
F[59,x] = F[57,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[72,x]
F[62,x] = F[2,x]+F[63,x]
F[63,x] = F[19,x]+F[48,x]+F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[4,x]+F[67,x]
F[67,x] = F[19,x]+F[27,x]+F[68,x]
F[68,x] = F[4,x]*F[7,x]
F[69,x] = F[39,x]+F[70,x]
F[70,x] = 2*F[19,x]+F[42,x]+F[71,x]
F[71,x] = F[4,x]*F[52,x]
F[72,x] = F[52,x]+F[73,x]
F[73,x] = F[57,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_{14}\! \left(x \right)+F_{7}\! \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_{4}\! \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_{60}\! \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_{35}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \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_{31}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \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_{39}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{69}\! \left(x \right) = F_{39}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{72}\! \left(x \right) = F_{52}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{57}\! \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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_60(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_35(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_1(x) + F_4(x))
Eq(F_26(x), F_19(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(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))
Eq(F_31(x), F_11(x)*F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_15(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_2(x)*F_4(x))
Eq(F_41(x), 2*F_19(x) + F_42(x) + F_50(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_2(x) + F_47(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_59(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_49(x) + F_55(x))
Eq(F_55(x), F_52(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_39(x))
Eq(F_59(x), F_57(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_72(x))
Eq(F_62(x), F_2(x) + F_63(x))
Eq(F_63(x), F_19(x) + F_48(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_4(x) + F_67(x))
Eq(F_67(x), F_19(x) + F_27(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_7(x))
Eq(F_69(x), F_39(x) + F_70(x))
Eq(F_70(x), 2*F_19(x) + F_42(x) + F_71(x))
Eq(F_71(x), F_4(x)*F_52(x))
Eq(F_72(x), F_52(x) + F_73(x))
Eq(F_73(x), F_57(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 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, 2, 3], "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, 3, 2], "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, 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, 2, 3], "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, 3, 2], "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, 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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [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, 0, 1], "pos": [[0, 1], [0, 0], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [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, 0, 1], "pos": [[1, 2], [1, 0], [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, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "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": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [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, 0, 1], "pos": [[0, 1], [0, 0], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 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, 0, 1], "pos": [[0, 1], [0, 0], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"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]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": 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