0132_0321_1203_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 146, 397, 1093, 3020, 8332, 22957, 63243, 174266, 480249, 1323480, 3647165, 10050539, 27696485, 76324044, 210328592, 579608874, 1597245244, 4401575777, 12129552899, 33425769470, 92112385924, 253836836328, 699505704186, 1927648633856, 5312078566694, 14638652607016, 40340169563315, 111166602827310, 306345107571204, 844204307298803, 2326398871244563, 6410926432525731, 17666780289138922, 48684870910568525, 134162344059758436, 369715154358244175, 1018835026475487372, 2807633928273986206, 7737080165436162772, 21321301500009609931, 58755743501939936572, 161914946630448560274, 446193825144522174932, 1229589569957966509610, 3388416480348457864171, 9337559885685694783467, 25731790978008760993755, 70909860289191659575038, 195408407076287585144332, 538492748404305245235195, 1483940452832288613661383, 4089338759115019094616313, 11269112217328522815429170, 31054627079666475954237508, 85578157751789119420264026, 235830269846819851631413817, 649884475631413642950670621, 1790905942401069680688763895, 4935252671501773672243856653, 13600222298056765919851823985, 37478536332016447685587278235, 103280715182940464888880213558, 284613732889759314749888509531, 784318512957231848324960156876, 2161369809958236806793281030245, 5956150949166283745826427957569, 16413541988883385350575906600024, 45231284922110312645455209003646, 124645194625922686769073402155364, 343488463130958026896812008241329, 946561354877375258713353074850543, 2608467225886689013337514533117500, 7188230571072124532265204409952053, 19808820379305942085967365107993305, 54587754377094436561284591434247728, 150429095265409202616130686676205639, 414541923561246067799660641892173672, 1142365485125491616202108515875982988, 3148050480383304570327129115968318056, 8675176163916483169751981951606477315, 23906440491964811366386488519291726297, 65879687766229526772871358905174366048, 181546611325707549093844958647847973874, 500293386343926616870364911777841065725, 1378673336790787606441875072938065682890, 3799251042409705000276613506223419230886, 10469708884666435612813159089656975697667, 28851687584229559974135141659817645931035, 79507451986283857105136797241827696186576, 219101045749800533543579618299772736506566, 603783256152363728912154647293509968602267, 1663863443291131610885885260393304284325441, 4585157885236236138599786048755170707986660, 12635455702397719329773770611531238241342561, 34819900383654798385959086507502777647105122
Generating function in Maple syntax:
(x^2+x-1)*(x-1)^2/(x^8+x^7-3*x^6-3*x^5+3*x^4+x^3-4*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{8}+x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**2*(x**2 + x - 1)/(x**8 + x**7 - 3*x**6 - 3*x**5 + 3*x**4 + x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^8+x^7-3*x^6-3*x^5+3*x^4+x^3-4*x^2+4*x-1)*F(x)-(x^2+x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}+x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+6)+238487874/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+6)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+5)+432773885/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+5)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+4)-471335374/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+4)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+3)-1186946263/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+3)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+2)-121101255/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+2)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n+1)+395394806/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n+1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n-1)+502849844/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n-1)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 1)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 2)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 3)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 4)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 5)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 6)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 7)^(-n)-251405927/3639954341*RootOf(_Z^8+_Z^7-3*_Z^6-3*_Z^5+3*_Z^4+_Z^3-4*_Z^2+4*_Z-1,index = 8)^(-n)
Explicit closed form in latex syntax:
\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +5}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +4}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +3}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +2}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}-3 \textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}+\textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{3639954341}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 146
a(7) = 397
a(n+8) = a(n)+a(n+1)-3*a(n+2)-3*a(n+3)+3*a(n+4)+a(n+5)-4*a(n+6)+4*a(n+7), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 146
a \! \left(7\right) = 397
a \! \left(n +8\right) = a \! \left(n \right)+a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)-4 a \! \left(n +6\right)+4 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19426/
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[21,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[56,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[29,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[19,x]
F[29,x] = F[30,x]+F[43,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[34,x]+F[37,x]
F[37,x] = 2*F[15,x]+F[38,x]+F[40,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[34,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[34,x]+F[42,x]
F[42,x] = F[40,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[47,x]+F[55,x]
F[47,x] = F[15,x]+F[35,x]+F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[52,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[18,x]
F[52,x] = F[34,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[46,x]
F[55,x] = F[54,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[33,x]+F[58,x]
F[58,x] = F[47,x]+F[59,x]
F[59,x] = 2*F[15,x]+F[38,x]+F[54,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_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{56}\! \left(x \right)
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_{29}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \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_{19}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{43}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{55}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \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_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{18}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{55}\! \left(x \right) = F_{54}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{33}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{47}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{54}\! \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_21(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_56(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_29(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_19(x))
Eq(F_29(x), F_30(x) + F_43(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_34(x) + F_37(x))
Eq(F_37(x), 2*F_15(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_34(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_34(x) + F_42(x))
Eq(F_42(x), F_40(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_47(x) + F_55(x))
Eq(F_47(x), F_15(x) + F_35(x) + F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_52(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_18(x))
Eq(F_52(x), F_34(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_46(x))
Eq(F_55(x), F_54(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_33(x) + F_58(x))
Eq(F_58(x), F_47(x) + F_59(x))
Eq(F_59(x), 2*F_15(x) + F_38(x) + F_54(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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