0123_0132_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 60, 173, 502, 1464, 4272, 12455, 36298, 105783, 308291, 898472, 2618443, 7630957, 22238943, 64811049, 188879049, 550450657, 1604179386, 4675062665, 13624542468, 39706024767, 115715326931, 337229347163, 982787978700, 2864140436682, 8346968636576, 24325582816597, 70892081305269, 206600895407352, 602097289212784, 1754693003392643, 5113704365169055, 14902876049291891, 43431473288189652, 126572405604072499, 368870145253351933, 1074998799380965798, 3132870560387574436, 9130129218556540196, 26607948825443742222, 77543583858425204050, 225985378920247642951, 658589517592672432501, 1919328386444048575847, 5593501500715181850949, 16301149537244652685080, 47506463742190023910258, 138448156194847524207027, 403479662426022139051876, 1175861365479675154724458, 3426814483075053339154901, 9986770418826150381366528, 29104459517996222386847173, 84819168591065482844718858, 247188626060247839200369211, 720382171489350154502342356, 2099410807329082079146037278, 6118315961120794216633644643, 17830617080479633507717834081, 51963793222024337924917074638, 151438157963553087968948428447, 441336443423339514516062977080, 1286187437253694809131697158280, 3748337914080681456253699509082, 10923786620194927434303817200628, 31835207192843607888337716197557, 92777390500986967282805542348362, 270381283716211566982725287629260, 787972567338470153777957711621456, 2296389595996107060464404189333493, 6692371530662685853607272031098532, 19503588059497699914464874014509513, 56839334972921718899797952833344529, 165646956360459886565329972559143776, 482745165202161695810531940663678259, 1406864935199557862612124373577014923, 4100028521394279371204660401547121183, 11948719067237325051041171012673266565, 34822169309985428804883587320510019899, 101482298531741585147973334225131250493, 295750009817232426746989642401047959916, 861904682613538319949289686213265416797, 2511850066785218254293851436923241941802, 7320288293221764890463448643807375707611, 21333526791454655656524391013339610504303, 62172328073900071931815410348971475271005, 181188905890468359889266481200231133613066, 528039091261997992046011789850842001765150, 1538865089617302846186287134681107651787506, 4484716762888077036861130374341855492448353, 13069816567436131904409245778960589829225425, 38089385381034166849226357552744865782510600, 111003951066892348310960130904278926903821188, 323498975612152001506182219509060740073407681, 942773533881216352658382266578670406365325003, 2747526277339745166668368372414170371411316895, 8007119815503341411483055304123954735128123348
Generating function in Maple syntax:
(x^3+x^2+x-1)^2/(2*x^8+4*x^7+5*x^6+x^5+x^4-3*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}+x^{2}+x -1\right)^{2}}{2 x^{8}+4 x^{7}+5 x^{6}+x^{5}+x^{4}-3 x +1}
Generating function in sympy syntax:
(x**3 + x**2 + x - 1)**2/(2*x**8 + 4*x**7 + 5*x**6 + x**5 + x**4 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^8+4*x^7+5*x^6+x^5+x^4-3*x+1)*F(x)-(x^3+x^2+x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{8}+4 x^{7}+5 x^{6}+x^{5}+x^{4}-3 x +1\right) F \! \left(x \right)-\left(x^{3}+x^{2}+x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+6)-837853372/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+6)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+5)-2404648612/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+5)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+4)-4003537778/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+4)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+3)-84921184/276031811*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+3)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+2)-2376472456/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+2)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n+1)-526591371/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n+1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n-1)+904645197/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n-1)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 1)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 2)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 3)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 4)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 5)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 6)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 7)^(-n)+1710328326/12973495117*RootOf(2*_Z^8+4*_Z^7+5*_Z^6+_Z^5+_Z^4-3*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{12973495117}-\frac{837853372 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{12973495117}-\frac{2404648612 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{12973495117}-\frac{4003537778 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{12973495117}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{276031811}-\frac{84921184 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{276031811}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{12973495117}-\frac{2376472456 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{12973495117}-\frac{526591371 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{12973495117}+\frac{904645197 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{12973495117}+\frac{1710328326 \mathit{RootOf}\! \left(2 \textit{\_Z}^{8}+4 \textit{\_Z}^{7}+5 \textit{\_Z}^{6}+\textit{\_Z}^{5}+\textit{\_Z}^{4}-3 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{12973495117}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 173
a(7) = 502
a(n+4) = -2*a(n)-4*a(n+1)-5*a(n+2)-a(n+3)+3*a(n+7)-a(n+8), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 173
a \! \left(7\right) = 502
a \! \left(n +4\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-5 a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +7\right)-a \! \left(n +8\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14858/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[19,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[4,x]*F[7,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[13,x]+F[22,x]+F[76,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[41,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[13,x]+F[14,x]+F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]+F[40,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[18,x]+F[7,x]
F[33,x] = F[29,x]+F[34,x]
F[34,x] = 2*F[13,x]+F[35,x]+F[39,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[18,x]
F[38,x] = F[39,x]
F[39,x] = F[29,x]*F[4,x]
F[40,x] = 2*F[13,x]+F[35,x]+F[39,x]
F[41,x] = F[42,x]+F[56,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[2,x]*F[4,x]
F[48,x] = F[42,x]+F[49,x]
F[49,x] = 2*F[13,x]+F[50,x]+F[55,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[46,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[42,x]
F[56,x] = 2*F[13,x]+F[57,x]+F[61,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[46,x]
F[60,x] = F[54,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[48,x]+F[63,x]
F[63,x] = F[64,x]+F[75,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[42,x]+F[54,x]
F[68,x] = F[64,x]+F[69,x]
F[69,x] = 3*F[13,x]+F[70,x]+F[74,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[54,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[64,x]
F[75,x] = 3*F[13,x]+F[70,x]+F[74,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[85,x]
F[78,x] = F[2,x]+F[79,x]
F[79,x] = F[13,x]+F[47,x]+F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[12,x]+F[4,x]
F[83,x] = F[46,x]+F[84,x]
F[84,x] = 2*F[13,x]+F[55,x]+F[57,x]
F[85,x] = F[42,x]+F[86,x]
F[86,x] = 2*F[13,x]+F[50,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{76}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{29}\! \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_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{56}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)
F_{53}\! \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_{42}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \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_{46}\! \left(x \right)
F_{60}\! \left(x \right) = F_{54}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{48}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{75}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \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_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{42}\! \left(x \right)+F_{54}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{54}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{75}\! \left(x \right) = 3 F_{13}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{85}\! \left(x \right)
F_{78}\! \left(x \right) = F_{2}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{13}\! \left(x \right)+F_{47}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{83}\! \left(x \right) = F_{46}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{85}\! \left(x \right) = F_{42}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{50}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_19(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_4(x)*F_7(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_13(x) + F_22(x) + F_76(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_41(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_13(x) + F_14(x) + F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_40(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_18(x) + F_7(x))
Eq(F_33(x), F_29(x) + F_34(x))
Eq(F_34(x), 2*F_13(x) + F_35(x) + F_39(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_18(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_29(x)*F_4(x))
Eq(F_40(x), 2*F_13(x) + F_35(x) + F_39(x))
Eq(F_41(x), F_42(x) + F_56(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_2(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_2(x)*F_4(x))
Eq(F_48(x), F_42(x) + F_49(x))
Eq(F_49(x), 2*F_13(x) + F_50(x) + F_55(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_46(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_42(x))
Eq(F_56(x), 2*F_13(x) + F_57(x) + F_61(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_46(x))
Eq(F_60(x), F_54(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_48(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_75(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_42(x) + F_54(x))
Eq(F_68(x), F_64(x) + F_69(x))
Eq(F_69(x), 3*F_13(x) + F_70(x) + F_74(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_54(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_64(x))
Eq(F_75(x), 3*F_13(x) + F_70(x) + F_74(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_85(x))
Eq(F_78(x), F_2(x) + F_79(x))
Eq(F_79(x), F_13(x) + F_47(x) + F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_12(x) + F_4(x))
Eq(F_83(x), F_46(x) + F_84(x))
Eq(F_84(x), 2*F_13(x) + F_55(x) + F_57(x))
Eq(F_85(x), F_42(x) + F_86(x))
Eq(F_86(x), 2*F_13(x) + F_50(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": 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