0321_1302_2031_2103_2301
Counting sequence:
1, 1, 2, 6, 19, 54, 140, 341, 798, 1822, 4099, 9142, 20288, 44897, 99202, 219006, 483275, 1066174, 2351836, 5187493, 11441774, 25236070, 55660395, 122763406, 270763808, 597189025, 1317142562, 2905050134, 6407290595, 14131725158, 31168501964, 68744296149, 151620319198, 334409142222, 737562582579, 1626745486470, 3587900117408, 7913362819777, 17453471128546, 38494842377166, 84903047576923, 187259566285358, 413013974951004, 910930997482213, 2009121561253230, 4431257097461078, 9773445192408155, 21556011946073502, 47543280989612224, 104860007171636929, 231276026289351874, 510095333568320678, 1125050674308283187, 2481377374905923350, 5472850083380172684, 12070750841068634069, 26622879057043197214, 58718608197466573054, 129507967236001786339, 285638813529046776278, 629996235255560132224, 1389500437747122057633, 3064639689023290898626, 6759275613302141936798, 14908051664351405938795, 32880743017726102784030, 72520761648754347512924, 159949574961860100972965, 352779892941446304738542, 778080547531646956998854, 1716110670025154014979787, 3785001232991754334707502, 8348083013515155626423520, 18412276697055465267836769, 40609554627102684870391266, 89567192267720525367216566, 197546661232496516002280707, 435702877092095716874963782, 960972946451911959117155532, 2119492554136320434236603477, 4674687985364736585348182750, 10310348917181385129813533358, 22740190388499090693863682835, 50155068762362917973075561382, 110620486441907221075964669408, 243981163272313532845793035265, 538117395306989983664661645858, 1186855277055887188405287975406, 2617691717384087909656369000699, 5773500830075165802977399662222, 12733856937206218794360087315164, 28085405591796525498376543646693, 61944312013668216799730486971630, 136622480964542652393821061274806, 301330367520881830286018666213051, 664605047055431877371767819414846, 1465832575075406407137356700121984, 3232995517671694644560732066474881, 7130596082398821166493231952382850, 15727024739873048740123820604906310, 34687044997417792124808373276306515
Generating function in Maple syntax:
(x^6-x^5+4*x^4-3*x^3+6*x^2-4*x+1)/(x^3+2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{6}-x^{5}+4 x^{4}-3 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**6 - x**5 + 4*x**4 - 3*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+2*x-1)*(x-1)^3*F(x)-x^6+x^5-4*x^4+3*x^3-6*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{6}+x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n < 0,23/1888*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*((-1888/23*(n^2+3/2)*(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n-37/69*59^(1/2)*3^(n+1/6)*2^(4*n+1/3)*((-3*I*59^(1/2)+9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)+48)^(-n)*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+(59^(1/2)*3^(1/6)*2^(1/3)-3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)*(9+59^(1/2)*3^(1/2))^(2/3)*(-27648*(108+12*59^(1/2)*3^(1/2))^(1/3)-27648*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-3456*I*59^(1/2)-10368)*18^(1/3)+31104*I*2^(1/3)*3^(1/6)+3456*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n-(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(2^(2/3)*((I*3^(1/3)+1/3*3^(5/6))*59^(1/2)-59/69*I*3^(5/6)-59/69*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)-944/23+37/138*((I*3^(2/3)-3^(1/6))*59^(1/2)-885/37*I*3^(1/6)+295/37*3^(2/3))*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3)))*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(944/23*((-3*I*59^(1/2)+9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)+48)^(-n)*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+(59^(1/2)*3^(1/6)*2^(1/3)-3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)*(-4*(108+12*59^(1/2)*3^(1/2))^(1/3)+4*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/2*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*331776^n+2/3*((-3*I*59^(1/2)+9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)+48)^(-n)*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+(59^(1/2)*3^(1/6)*2^(1/3)-3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)*(295/46*3^(n+2/3)*2^(4*n+1/3)*(9+59^(1/2)*3^(1/2))^(2/3)+2^(4*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*(59^(1/2)*3^(n+5/6)-59/23*3^(n+1/3)))*(-27648*(108+12*59^(1/2)*3^(1/2))^(1/3)+27648*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((3456*I*59^(1/2)-10368)*18^(1/3)-31104*I*2^(1/3)*3^(1/6)+3456*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(((I*3^(1/3)-1/3*3^(5/6))*59^(1/2)+59/69*3^(1/3)-59/69*I*3^(5/6))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+944/23+37/138*2^(1/3)*((I*3^(2/3)+3^(1/6))*59^(1/2)-885/37*I*3^(1/6)-295/37*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3)))),n = 0,1,0 < n,23/1888*(-(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*((1888/23*n^2+2832/23)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+2^(2/3)*((I*3^(1/3)+1/3*3^(5/6))*59^(1/2)-59/69*I*3^(5/6)-59/69*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)-944/23+37/138*((I*3^(2/3)-3^(1/6))*59^(1/2)-885/37*I*3^(1/6)+295/37*3^(2/3))*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3))*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(((I*3^(1/3)-1/3*3^(5/6))*59^(1/2)+59/69*3^(1/3)-59/69*I*3^(5/6))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+944/23+37/138*2^(1/3)*((I*3^(2/3)+3^(1/6))*59^(1/2)-885/37*I*3^(1/6)-295/37*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3))*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+2/3*((-3*I*59^(1/2)+9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)+48)^(-n)*((3^(1/6)*59^(1/2)*(9+59^(1/2)*3^(1/2))^(1/3)-3*(81+9*59^(1/2)*3^(1/2))^(1/3)-8*6^(1/3))^(-n)*(18+2*59^(1/2)*3^(1/2))^(-1/3*n)*(1/3456*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/384*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/72*I*3^(1/2)-1/72)^(-n)*(2^(2/3)*(59^(1/2)*3^(5/6)-59/23*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)+295/46*2^(1/3)*3^(2/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+8*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(-4*(108+12*59^(1/2)*3^(1/2))^(1/3)-4*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/2*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n-37/46*(59^(1/2)*3^(1/6)*2^(1/3)*(-288*(108+12*59^(1/2)*3^(1/2))^(1/3)-288*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-36*I*59^(1/2)-108)*18^(1/3)+324*I*2^(1/3)*3^(1/6)+36*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(9+59^(1/2)*3^(1/2))^(2/3)*(3^(1/6)*59^(1/2)*(9+59^(1/2)*3^(1/2))^(1/3)-3*(81+9*59^(1/2)*3^(1/2))^(1/3)-8*6^(1/3))^(-n)*((48*I*59^(1/2)-144)*(81+9*59^(1/2)*3^(1/2))^(1/3)-432*3^(1/6)*(I-1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)+384*I*3^(5/6)*2^(1/3)-384*6^(1/3))^n-2832/37*(-27648*(108+12*59^(1/2)*3^(1/2))^(1/3)-27648*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-3456*I*59^(1/2)-10368)*18^(1/3)+31104*I*2^(1/3)*3^(1/6)+3456*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(-8*(108+12*59^(1/2)*3^(1/2))^(1/3)+8*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(2*3^(1/6)*59^(1/2)*(9+59^(1/2)*3^(1/2))^(1/3)-6*(81+9*59^(1/2)*3^(1/2))^(1/3)-16*6^(1/3))^(-n)*(18+2*59^(1/2)*3^(1/2))^(-1/3*n))*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)))*(108+12*59^(1/2)*3^(1/2))^(-2/3*n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}\frac{23 \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(-\frac{1888 \left(n^{2}+\frac{3}{2}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}}{23}-\frac{37 \sqrt{59}\, 3^{n +\frac{1}{6}} 2^{4 n +\frac{1}{3}} \left(\left(-3 \,\mathrm{I} \sqrt{59}+9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}+48\right)^{-n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{69}-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}-\frac{59 \,3^{\frac{1}{3}}}{69}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{944}{23}+\frac{37 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}+\frac{295 \,3^{\frac{2}{3}}}{37}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right)\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(\frac{944 \left(\left(-3 \,\mathrm{I} \sqrt{59}+9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}+48\right)^{-n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-4 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n} 331776^{n}}{23}+\frac{2 \left(\left(-3 \,\mathrm{I} \sqrt{59}+9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}+48\right)^{-n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{295 \,3^{n +\frac{2}{3}} 2^{4 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{46}+2^{4 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{59}\, 3^{n +\frac{5}{6}}-\frac{59 \,3^{n +\frac{1}{3}}}{23}\right)\right) \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}-31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{3}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}+\frac{59 \,3^{\frac{1}{3}}}{69}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{944}{23}+\frac{37 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}-\frac{295 \,3^{\frac{2}{3}}}{37}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right)\right)\right)}{1888} & n <0 \\ 1 & n =0 \\ \frac{23 \left(-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\frac{1888 n^{2}}{23}+\frac{2832}{23}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}-\frac{59 \,3^{\frac{1}{3}}}{69}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{944}{23}+\frac{37 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}+\frac{295 \,3^{\frac{2}{3}}}{37}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}+\frac{59 \,3^{\frac{1}{3}}}{69}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{944}{23}+\frac{37 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}-\frac{295 \,3^{\frac{2}{3}}}{37}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{2 \left(\left(-3 \,\mathrm{I} \sqrt{59}+9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}+48\right)^{-n} \left(\left(3^{\frac{1}{6}} \sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}} \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{59 \,3^{\frac{1}{3}}}{23}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{295 \,2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{46}\right) \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-4 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-4 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n}-\frac{37 \left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(-288 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-288 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-36 \,\mathrm{I} \sqrt{59}-108\right) 18^{\frac{1}{3}}+324 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+36 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(3^{\frac{1}{6}} \sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{-n} \left(\left(48 \,\mathrm{I} \sqrt{59}-144\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-432 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+384 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-384 \,6^{\frac{1}{3}}\right)^{n}-\frac{2832 \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(2 \,3^{\frac{1}{6}} \sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-6 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-16 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}}}{37}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}}{46}\right)}{3}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{1888} & 0<n \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 140
a(n) = -2*n^2-2*a(n+2)+a(n+3)-2*n-2, n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 140
a \! \left(n \right) = -2 n^{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-2 n -2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21099/
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[24,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[7,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[4,x]
F[22,x] = F[15,x]+F[19,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[61,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[32,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[28,x]
F[32,x] = F[33,x]+F[56,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[1,x]+F[37,x]
F[40,x] = F[34,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[37,x]+F[42,x]
F[45,x] = 2*F[15,x]+F[46,x]+F[49,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[40,x]+F[48,x]
F[48,x] = F[46,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[53,x]
F[53,x] = F[37,x]*F[4,x]
F[54,x] = 2*F[15,x]+F[49,x]+F[55,x]
F[55,x] = F[4,x]*F[40,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[40,x]+F[60,x]
F[60,x] = F[57,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[67,x]
F[63,x] = F[37,x]+F[64,x]
F[64,x] = F[15,x]+F[61,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[40,x]+F[7,x]
F[67,x] = F[42,x]+F[68,x]
F[68,x] = 2*F[15,x]+F[46,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[52,x]+F[72,x]
F[72,x] = 2*F[15,x]+F[55,x]+F[69,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_{24}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{61}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \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_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \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_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{40}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{37}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \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_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{67}\! \left(x \right) = F_{42}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{52}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{69}\! \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_24(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_7(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_4(x))
Eq(F_22(x), F_15(x) + F_19(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_61(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_32(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_28(x))
Eq(F_32(x), F_33(x) + F_56(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_1(x) + F_37(x))
Eq(F_40(x), F_34(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_37(x) + F_42(x))
Eq(F_45(x), 2*F_15(x) + F_46(x) + F_49(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_40(x) + F_48(x))
Eq(F_48(x), F_46(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_37(x)*F_4(x))
Eq(F_54(x), 2*F_15(x) + F_49(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_40(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_40(x) + F_60(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_67(x))
Eq(F_63(x), F_37(x) + F_64(x))
Eq(F_64(x), F_15(x) + F_61(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_40(x) + F_7(x))
Eq(F_67(x), F_42(x) + F_68(x))
Eq(F_68(x), 2*F_15(x) + F_46(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_52(x) + F_72(x))
Eq(F_72(x), 2*F_15(x) + F_55(x) + F_69(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, 3, 2, 1], "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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 3, 2, 1], "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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 3, 2, 1], "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, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"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, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": 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