02341_02413_02431_03241_03412_03421_04123_04132_04213_04231_04312_04321_40123_40132_40213_40231_40312_40321

Counting sequence:
1, 1, 2, 6, 24, 102, 433, 1825, 7657, 32058, 134107, 560843, 2345284, 9807156, 41010189, 171491412, 717124043, 2998795586, 12540063470, 52438796274, 219283384148, 916977630673, 3834526613177, 16034845108755, 67052933200008, 280395339838443, 1172529564295423, 4903168432137886, 20503585926005446, 85739872419232401, 358538538040864024, 1499301079346738537, 6269629309086572824, 26217717184963398486, 109634662673755250574, 458459414097447626682, 1917140339091882639347, 8016908294944783787357, 33524316034160554452539, 140188676757965564198562, 586227175257608115201065, 2451426955144483772453419, 10251135344874067103496338, 42867186247747170845452152, 179257769503346512208627535, 749602452132104499027369888, 3134613566816550172593864493, 13108017703681079758892513650, 54813814991081544331885201348, 229215002740861201329044944266, 958508680522262959582489313506, 4008197018740561763056352172413, 16761082781521638934517245710787, 70089842065022788676214117410367, 293094785386762079857220808060858, 1225634852211793064722926687274515, 5125238884663115982475566518166821, 21432218231603963811956896147790078, 89623135362848543538318642550171508, 374777183839189756860206244587511645, 1567206245996363908463421120235862450, 6553588434412004266624906382842777237, 27405149435423085201968019247026538082, 114600149688108182926089075924968299590, 479223597721427627886286205937432907596, 2003969953251287413002598782762217762410, 8380003807467771231248961289586417641465, 35042673019742906294331253720767886978993, 146537992175409898599455848929875422443829, 612778115947447038180480068305269167385006, 2562455058989909347684445148398803486621723, 10715421713764508755815972437100871859630987, 44808693171413809664316051875678948837256664, 187376571577277272782463138040939314690285848, 783552857516793286814985512510043140440872741, 3276584021975910071184045347844082884886745220, 13701695744039494649944927274538472361477599751, 57296399238685620090347365579849571753677656806, 239596428576876690600526893223847624607510981230, 1001920702689366378133354870133203614929009423874, 4189733129329433547971799317397466558873136599448, 17520212575588404653237325533290403390235767567557, 73264296130225043462710683696738964456414746680369, 306369404155305409910178239390101512838299489729219, 1281145343096447139567812844366939790854600254693172, 5357367177910771895520231996621716490041964780002463, 22402909422896705417429477297311509059165365479801295, 93682276003011919465207477124647343568784973705414302, 391751297629882343877201752408305948894539744408570682, 1638186920114564388540240149351518637076039212780376645, 6850408413375312597086357342268121709861824474130707264, 28646361934547380451174369800998004657775415633598722137, 119790529639495560191328508817754462150369667422998505988, 500928216438021982605867093515293101868557600511975908474, 2094732186082973755237409730570234240307287090856358797758, 8759544356700963320339037354309078627978069398980845507258, 36629798237115730944350360885200586648720202299080029224135, 153174875798806531087017729746359926450445223679158272120481, 640531580984959095782566400789384321104598362225651391521523, 2678511760492564624643925415756115871628097108800059160650258, 11200736176137811820975822089461437451504887387869108444561781

Generating function in Maple syntax:
(x-1)*(3*x^6-6*x^5+4*x^4-12*x^3+15*x^2-7*x+1)/(9*x^7-12*x^6+14*x^5-32*x^4+44*x^3-29*x^2+9*x-1)

Generating function in latex syntax:
\frac{\left(x -1\right) \left(3 x^{6}-6 x^{5}+4 x^{4}-12 x^{3}+15 x^{2}-7 x +1\right)}{9 x^{7}-12 x^{6}+14 x^{5}-32 x^{4}+44 x^{3}-29 x^{2}+9 x -1}

Generating function in sympy syntax:
(x - 1)*(3*x**6 - 6*x**5 + 4*x**4 - 12*x**3 + 15*x**2 - 7*x + 1)/(9*x**7 - 12*x**6 + 14*x**5 - 32*x**4 + 44*x**3 - 29*x**2 + 9*x - 1)

Implicit equation for the generating function in Maple syntax:
(9*x^7-12*x^6+14*x^5-32*x^4+44*x^3-29*x^2+9*x-1)*F(x)-(x-1)*(3*x^6-6*x^5+4*x^4-12*x^3+15*x^2-7*x+1) = 0

Implicit equation for the generating function in latex syntax:
\left(9 x^{7}-12 x^{6}+14 x^{5}-32 x^{4}+44 x^{3}-29 x^{2}+9 x -1\right) F \! \left(x \right)-\left(x -1\right) \left(3 x^{6}-6 x^{5}+4 x^{4}-12 x^{3}+15 x^{2}-7 x +1\right) = 0

Explicit closed form in Maple syntax:
-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n+5)-1336476348/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n+5)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n+4)+511619211/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n+4)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n+3)-1394874942/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n+3)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n+2)+3496220935/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n+2)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n+1)-2996478561/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n+1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n-1)-108845404/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n-1)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 1)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 2)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 3)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 4)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 5)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 6)^(-n)+1092174886/1242744671*RootOf(9*_Z^7-12*_Z^6+14*_Z^5-32*_Z^4+44*_Z^3-29*_Z^2+9*_Z-1,index = 7)^(-n)+piecewise(n = 0,1/3,0)

Explicit closed form in latex syntax:
-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{1242744671}-\frac{1336476348 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{1242744671}+\frac{511619211 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{1242744671}-\frac{1394874942 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{1242744671}+\frac{3496220935 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{1242744671}-\frac{2996478561 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{1242744671}-\frac{108845404 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{1242744671}+\frac{1092174886 \mathit{RootOf}\! \left(9 \textit{\_Z}^{7}-12 \textit{\_Z}^{6}+14 \textit{\_Z}^{5}-32 \textit{\_Z}^{4}+44 \textit{\_Z}^{3}-29 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{1242744671}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 102
a(6) = 433
a(7) = 1825
a(n+7) = 9*a(n)-12*a(n+1)+14*a(n+2)-32*a(n+3)+44*a(n+4)-29*a(n+5)+9*a(n+6), n >= 8

Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 102
a(6) = 433
a(7) = 1825
a{\left(n + 7 \right)} = 9 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 14 a{\left(n + 2 \right)} - 32 a{\left(n + 3 \right)} + 44 a{\left(n + 4 \right)} - 29 a{\left(n + 5 \right)} + 9 a{\left(n + 6 \right)}, \quad n \geq 8

Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25460/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[2,x]+F[6,x]
F[6,x] = F[7,x]+F[73,x]+F[8,x]
F[7,x] = 0
F[8,x] = F[16,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[2,x]+F[6,x]
F[11,x] = F[12,x]+F[25,x]
F[12,x] = F[13,x]+F[17,x]+F[7,x]
F[13,x] = F[14,x]*F[16,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]+F[2,x]
F[16,x] = x
F[17,x] = F[16,x]*F[18,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[2,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[2,x]+F[21,x]
F[25,x] = F[26,x]+F[29,x]+F[30,x]+F[7,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[25,x]+F[6,x]
F[29,x] = 0
F[30,x] = F[16,x]*F[31,x]
F[31,x] = F[32,x]+F[42,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[16,x]*F[35,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[2,x]+F[33,x]
F[37,x] = F[21,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[33,x]+F[38,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[16,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[43,x]+F[47,x]
F[47,x] = F[48,x]+F[65,x]+F[7,x]
F[48,x] = F[16,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[2,x]+F[47,x]
F[51,x] = F[12,x]+F[52,x]
F[52,x] = F[53,x]+F[56,x]+F[57,x]+F[7,x]
F[53,x] = F[16,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[47,x]+F[52,x]
F[56,x] = 0
F[57,x] = F[16,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[33,x]
F[60,x] = F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[16,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[33,x]+F[61,x]
F[65,x] = F[16,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[2,x]+F[21,x]
F[68,x] = F[33,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[16,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[6,x]+F[69,x]
F[73,x] = F[16,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[12,x]+F[2,x]
F[76,x] = F[47,x]+F[77,x]
F[77,x] = F[30,x]+F[7,x]+F[78,x]+F[81,x]
F[78,x] = F[16,x]*F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[6,x]+F[77,x]
F[81,x] = 0
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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)+F_{73}\! \left(x \right)+F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = 0
F_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{25}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{65}\! \left(x \right)+F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)+F_{7}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{16}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{33}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{33}\! \left(x \right)+F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = F_{16}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{68}\! \left(x \right) = F_{33}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{16}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{6}\! \left(x \right)+F_{69}\! \left(x \right)
F_{73}\! \left(x \right) = F_{16}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right)+F_{2}\! \left(x \right)
F_{76}\! \left(x \right) = F_{47}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)+F_{78}\! \left(x \right)+F_{81}\! \left(x \right)
F_{78}\! \left(x \right) = F_{16}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{6}\! \left(x \right)+F_{77}\! \left(x \right)
F_{81}\! \left(x \right) = 0
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_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_2(x) + F_6(x))
Eq(F_6(x), F_7(x) + F_73(x) + F_8(x))
Eq(F_7(x), 0)
Eq(F_8(x), F_16(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_2(x) + F_6(x))
Eq(F_11(x), F_12(x) + F_25(x))
Eq(F_12(x), F_13(x) + F_17(x) + F_7(x))
Eq(F_13(x), F_14(x)*F_16(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x) + F_2(x))
Eq(F_16(x), x)
Eq(F_17(x), F_16(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_2(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_2(x) + F_21(x))
Eq(F_25(x), F_26(x) + F_29(x) + F_30(x) + F_7(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_25(x) + F_6(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_16(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_42(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_16(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_2(x) + F_33(x))
Eq(F_37(x), F_21(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_33(x) + F_38(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_16(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_43(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_65(x) + F_7(x))
Eq(F_48(x), F_16(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_2(x) + F_47(x))
Eq(F_51(x), F_12(x) + F_52(x))
Eq(F_52(x), F_53(x) + F_56(x) + F_57(x) + F_7(x))
Eq(F_53(x), F_16(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_47(x) + F_52(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_16(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_33(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_16(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_33(x) + F_61(x))
Eq(F_65(x), F_16(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_2(x) + F_21(x))
Eq(F_68(x), F_33(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_16(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_6(x) + F_69(x))
Eq(F_73(x), F_16(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_12(x) + F_2(x))
Eq(F_76(x), F_47(x) + F_77(x))
Eq(F_77(x), F_30(x) + F_7(x) + F_78(x) + F_81(x))
Eq(F_78(x), F_16(x)*F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_6(x) + F_77(x))
Eq(F_81(x), 0)
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [], "assumptions": []}, "rules": [{"class_module": "comb_spec_searcher.strategies.rule", "rule_class": "Rule", "comb_class": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [], "assumptions": []}, "children": [{"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, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]], "assumptions": []}], "strategy": {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "RequirementInsertionStrategy", "ignore_parent": true, "gps": [{"patt": [0], "pos": [[0, 0]]}]}}, {"class_module": "comb_spec_searcher.strategies.rule", "rule_class": "Rule", "comb_class": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 2], [0, 0], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 2], [0, 0], [0, 0], [0, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 2], [0, 0], [0, 2], [0, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 2], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], 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