0132_0231_1023_1302_2103

Counting sequence:
1, 1, 2, 6, 19, 53, 143, 388, 1061, 2905, 7948, 21738, 59456, 162628, 444832, 1216725, 3328033, 9102975, 24898861, 68104474, 186282366, 509527729, 1393682684, 3812062290, 10426920771, 28520173176, 78009634441, 213375389714, 583633768643, 1596380802664, 4366491118218, 11943418922079, 32668165739155, 89355406498344, 244408845425807, 668517844227742, 1828559466709406, 5001556431384552, 13680477551731654, 37419445048953451, 102351315038297622, 279955827147196880, 765747515059722318, 2094506346931064013, 5728986057489670562, 15670175120262491647, 42861753517216747083, 117237356983657523104, 320672785050490507641, 877118332566746404243, 2399132714687765340109, 6562213522366530052579, 17949255599532150573989, 49095594874997310575749, 134288434568433520713376, 367311643840051114959203, 1004686994334767466126220, 2748064139847902207435407, 7516626132617845689709775, 20559807028623652431424883, 56236090181463519695102600, 153819431986634102407939834, 420733688639145048203996472, 1150809325386682292489699485, 3147744378826840146989682707, 8609849134744101531896513466, 23550038758446362097806185156, 64415103777635425240293592931, 176191030394604696360555716323, 481925470440530038503080184755, 1318183783471636894791198752252, 3605554372171373916347080882783, 9862071202580402017052127923131, 26975171739316323923998264410744, 73783678440206202911867184966772, 201816368651068564995817328970004, 552017024856129999132369892535956, 1509901291791971887815241034243192, 4129948549230417548596705983047333, 11296417263838199960877229946898275, 30898458292542321250674821878269439, 84514824705722010615153964802853773, 231168672799539038386977450636263862, 632302740613532824113285935985690288, 1729502319434442875292801364957667681, 4730610957065807719736841069370433510, 12939375550782165658555397591370150256, 35392350198255836626528904264656243747, 96806715875887926406741331738317626350, 264789995187624711784307183657532630275, 724265263180215530403418297115697673224, 1981042263616547831943287189714012524857, 5418634096853618915083817167368808127286, 14821286761435741320114943635669540593080, 40539836670696059829088652962232095744439, 110886347706527262497226968634591912586809, 303301224609540295147712770835715045051656, 829602874946450922105308153996304647311781, 2269166341169359625522546323560355657673788, 6206723770368205316548858974751595184754248, 16976904364711140458135487980316485991855172

Generating function in Maple syntax:
(x-1)^3/(x^7+2*x^6-2*x^5+x^4+4*x^3-5*x^2+4*x-1)

Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{x^{7}+2 x^{6}-2 x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1}

Generating function in sympy syntax:
(x - 1)**3/(x**7 + 2*x**6 - 2*x**5 + x**4 + 4*x**3 - 5*x**2 + 4*x - 1)

Implicit equation for the generating function in Maple syntax:
(x^7+2*x^6-2*x^5+x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{7}+2 x^{6}-2 x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0

Explicit closed form in Maple syntax:
2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+5)+2050875/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+5)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+4)+4555427/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+4)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+3)-3542146/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+3)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+2)-1413135/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+2)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+1)+6917142/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n-1)+2516419/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n-1)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n)-4476334/13769453*RootOf(_Z^7+2*_Z^6-2*_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n)

Explicit closed form in latex syntax:
\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{13769453}+\frac{2050875 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{13769453}+\frac{4555427 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{13769453}-\frac{3542146 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{13769453}-\frac{1413135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{13769453}+\frac{6917142 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{13769453}+\frac{2516419 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{13769453}-\frac{4476334 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-2 \textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{13769453}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 143
a(n+7) = a(n)+2*a(n+1)-2*a(n+2)+a(n+3)+4*a(n+4)-5*a(n+5)+4*a(n+6), 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) = 143
a \! \left(n +7\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18781/
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[20,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[17,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[61,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[33,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[15,x]+F[22,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[30,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[34,x]
F[37,x] = 2*F[22,x]+F[38,x]+F[45,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]*F[4,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[40,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[60,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[2,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[50,x]
F[53,x] = F[47,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[41,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[57,x]
F[60,x] = F[45,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[74,x]
F[63,x] = F[2,x]+F[64,x]
F[64,x] = F[22,x]+F[65,x]+F[73,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[4,x]+F[68,x]
F[68,x] = F[15,x]+F[22,x]+F[69,x]
F[69,x] = F[4,x]*F[7,x]
F[70,x] = F[41,x]+F[71,x]
F[71,x] = 2*F[22,x]+F[38,x]+F[72,x]
F[72,x] = F[4,x]*F[47,x]
F[73,x] = F[4,x]*F[63,x]
F[74,x] = F[47,x]+F[75,x]
F[75,x] = F[55,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_{20}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{61}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \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_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \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_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \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_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{45}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{74}\! \left(x \right)
F_{63}\! \left(x \right) = F_{2}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{65}\! \left(x \right)+F_{73}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{70}\! \left(x \right) = F_{41}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{74}\! \left(x \right) = F_{47}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{55}\! \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_20(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_17(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_61(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_33(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_15(x) + F_22(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_30(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_34(x))
Eq(F_37(x), 2*F_22(x) + F_38(x) + F_45(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x)*F_4(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_40(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_60(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_2(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_50(x))
Eq(F_53(x), F_47(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_41(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_57(x))
Eq(F_60(x), F_45(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_74(x))
Eq(F_63(x), F_2(x) + F_64(x))
Eq(F_64(x), F_22(x) + F_65(x) + F_73(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_4(x) + F_68(x))
Eq(F_68(x), F_15(x) + F_22(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_7(x))
Eq(F_70(x), F_41(x) + F_71(x))
Eq(F_71(x), 2*F_22(x) + F_38(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_47(x))
Eq(F_73(x), F_4(x)*F_63(x))
Eq(F_74(x), F_47(x) + F_75(x))
Eq(F_75(x), F_55(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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