0123_0231_0312_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 417, 1078, 2712, 6675, 16142, 38484, 90695, 211730, 490454, 1128759, 2583720, 5887024, 13361281, 30223474, 68168632, 153368175, 344299430, 771447088, 1725621967, 3854263062, 8597413118, 19155328571, 42634681544, 94806299540, 210646975025, 467686727102, 1037695945576, 2301071250715, 5099893998542, 11297613483676, 25016557301159, 55373526768170, 122525703355830, 271030036685711, 599357745486712, 1325089485024904, 2928905588128305, 6472562084530458, 14300999979661736, 31592478198605335, 69780663784050054, 154108618152381592, 340302295712609999, 751370417627755086, 1658819778244863998, 3661882501876284723, 8083016720728220344, 17840615818397098764, 39374639336062072961, 86895345787635550342, 191757408183234570360, 423141657281663958371, 933683063509228959438, 2060132341518223477028, 4545423952951172891207, 10028566194677698702722, 22125335181405868807638, 48812235216827406360359, 107685318430461503134856, 237560535646586858503968, 524062433718517090225441, 1156068440284527617304066, 2550213925049705960552920, 5625523301486056746217215, 12409181078592896579514790, 27372708152908010059139648, 60379203748647098743615503, 133184116858577137824988614, 293773998435442373225607486, 647993313750292020227816811, 1429302970620681528346600168, 3152644392199846130050769092, 6953811003196065680593283089, 15337982787104975690061028238, 33830725586594123111228555208, 74619493416752963105161858795, 164585432101348204304607682574, 363018514750765136528889800172, 800690372841232445779833225255, 1766033877629711515098057671962, 3895221669701985005192572228662, 8591404511628796133100111858687, 18949384499654491135168549748696, 41795073866545341983270208452088, 92183718639788229515121602222417, 203321154569368448996373901120298, 448446048585557237637942304554056, 989093146971452700114854799061287, 2181542110833373839873780674712582, 4811599594894504898680898004927016, 10612430986044862460067439510819471, 23406681381491898685190237100168574, 51625516955015902119424527012907198, 113864574090351866399642803151929059
Generating function in Maple syntax:
-(5*x^6-7*x^5+10*x^4-15*x^3+14*x^2-6*x+1)/(2*x-1)/(x^3+2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{5 x^{6}-7 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-5*x**6 + 7*x**5 - 10*x**4 + 15*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^3+2*x-1)*(x-1)^3*F(x)+5*x^6-7*x^5+10*x^4-15*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+5 x^{6}-7 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
469/11328*(-11328/67*(-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*3^(1/6)*2^(1/3)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(1/81*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/9*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+16/27*I*3^(1/2)-16/27)^(-n)*(-2048/3*(108+12*59^(1/2)*3^(1/2))^(1/3)-2048/3*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/3*((-256*I*59^(1/2)-768)*18^(1/3)+2304*I*2^(1/3)*3^(1/6)+256*3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-((-5664/469*n^2-36816/469)*(-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)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-531/469*(I-469/1593*59^(1/2))*3^(1/2)*(108+12*59^(1/2)*3^(1/2))^(1/3)-5723/5628*3^(1/2)*(I+219/5723*59^(1/2))*(108+12*59^(1/2)*3^(1/2))^(2/3)+3^(1/3)*2^(2/3)*(I*59^(1/2)-531/469)*(9+59^(1/2)*3^(1/2))^(1/3)-17936/469+219/938*(I*59^(1/2)+5723/657)*2^(1/3)*3^(2/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(1/331776*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/36864*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(108+12*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*3^(1/6)*2^(1/3)+3^(1/6)*2^(1/3)*59^(1/2))*(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)+3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(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*3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(59^(1/2)*16^n*3^(1/2+n)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-219/1876*59^(1/2)*16^n*3^(1/2+n)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)+5723/938*(3^(n+2/3)*2^(4*n+1/3)*(9+59^(1/2)*3^(1/2))^(2/3)-18/97*3^(4/3+n)*2^(4*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3))*(108+12*59^(1/2)*3^(1/2))^(-2/3*n))*(384*(I*3^(5/6)-3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+96*2^(1/3)*(3^(1/6)*59^(1/2)-3*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3)+(-576*I-192*3^(1/2))*59^(1/2)+1728*I*3^(1/2)+1728)^(-n)+(-531/469*(469/1593*59^(1/2)+I)*3^(1/2)*(108+12*59^(1/2)*3^(1/2))^(1/3)-5723/5628*(-219/5723*59^(1/2)+I)*3^(1/2)*(108+12*59^(1/2)*3^(1/2))^(2/3)+219/938*(I*59^(1/2)-5723/657)*2^(1/3)*3^(2/3)*(9+59^(1/2)*3^(1/2))^(2/3)+3^(1/3)*(531/469+I*59^(1/2))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+17936/469+17936/469*((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)-3^(1/6)*2^(1/3)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))/(-16*(108+12*59^(1/2)*3^(1/2))^(1/3)+(2*3^(1/6)*2^(1/3)*59^(1/2)-6*18^(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*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/6912*I*3^(1/2)-1/6912)^(-n)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)))*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n)
Explicit closed form in latex syntax:
\frac{469 \left(-\frac{11328 \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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{81}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}+\frac{16 \,\mathrm{I} \sqrt{3}}{27}-\frac{16}{27}\right)^{-n} \left(-\frac{2048 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{2048 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-256 \,\mathrm{I} \sqrt{59}-768\right) 18^{\frac{1}{3}}+2304 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+256 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{67}-\left(\left(-\frac{5664 n^{2}}{469}-\frac{36816}{469}\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}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}-\frac{531 \left(\mathrm{I}-\frac{469 \sqrt{59}}{1593}\right) \sqrt{3}\, \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{469}-\frac{5723 \sqrt{3}\, \left(\mathrm{I}+\frac{219 \sqrt{59}}{5723}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{5628}+3^{\frac{1}{3}} 2^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}-\frac{531}{469}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{17936}{469}+\frac{219 \left(\mathrm{I} \sqrt{59}+\frac{5723}{657}\right) 2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{938}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\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}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(\frac{2 \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 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\sqrt{59}\, 16^{n} 3^{\frac{1}{2}+n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{219 \sqrt{59}\, 16^{n} 3^{\frac{1}{2}+n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{1876}+\frac{5723 \left(3^{n +\frac{2}{3}} 2^{4 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{18 \,3^{\frac{4}{3}+n} 2^{4 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{97}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{938}\right) \left(384 \left(\mathrm{I} \,3^{\frac{5}{6}}-3^{\frac{1}{3}}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+96 \,2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \sqrt{59}-3 \,3^{\frac{2}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-576 \,\mathrm{I}-192 \sqrt{3}\right) \sqrt{59}+1728 \,\mathrm{I} \sqrt{3}+1728\right)^{-n}}{3}+\left(-\frac{531 \left(\frac{469 \sqrt{59}}{1593}+\mathrm{I}\right) \sqrt{3}\, \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{469}-\frac{5723 \left(-\frac{219 \sqrt{59}}{5723}+\mathrm{I}\right) \sqrt{3}\, \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{5628}+\frac{219 \left(\mathrm{I} \sqrt{59}-\frac{5723}{657}\right) 2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{938}+3^{\frac{1}{3}} \left(\frac{531}{469}+\mathrm{I} \sqrt{59}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{17936}{469}+\frac{17936 \left(\frac{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}}-3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{-16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(2 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}-6 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}\right)^{n}}{469}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right)\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}}{11328}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 156
a(n+4) = n^2-2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)-n+4, 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) = 56
a \! \left(6\right) = 156
a \! \left(n +4\right) = n^{2}-2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-n +4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17042/
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[17,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[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[80,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[53,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[35,x]+F[40,x]+F[47,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[11,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[11,x]+F[43,x]
F[46,x] = F[14,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[13,x]+F[49,x]
F[49,x] = F[43,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[14,x]+F[50,x]
F[53,x] = F[34,x]+F[54,x]
F[54,x] = 2*F[35,x]+F[55,x]+F[73,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[69,x]
F[57,x] = F[58,x]+F[68,x]
F[58,x] = F[32,x]+F[35,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[11,x]+F[62,x]
F[62,x] = F[35,x]+F[40,x]+F[44,x]
F[63,x] = F[58,x]+F[64,x]
F[64,x] = 2*F[35,x]+F[55,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[31,x]+F[67,x]
F[67,x] = F[65,x]
F[68,x] = F[65,x]
F[69,x] = F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[58,x]+F[70,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[31,x]+F[70,x]
F[76,x] = F[67,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[70,x]+F[77,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[75,x]+F[82,x]
F[82,x] = F[18,x]+F[58,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_{17}\! \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_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \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_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{80}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{43}\! \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_{11}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{34}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{55}\! \left(x \right)+F_{73}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{69}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{68}\! \left(x \right)
F_{58}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{55}\! \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_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{65}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{31}\! \left(x \right)+F_{70}\! \left(x \right)
F_{76}\! \left(x \right) = F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{75}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{18}\! \left(x \right)+F_{58}\! \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_17(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))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_80(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_53(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_47(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_11(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_11(x) + F_43(x))
Eq(F_46(x), F_14(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_13(x) + F_49(x))
Eq(F_49(x), F_43(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_14(x) + F_50(x))
Eq(F_53(x), F_34(x) + F_54(x))
Eq(F_54(x), 2*F_35(x) + F_55(x) + F_73(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_69(x))
Eq(F_57(x), F_58(x) + F_68(x))
Eq(F_58(x), F_32(x) + F_35(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_11(x) + F_62(x))
Eq(F_62(x), F_35(x) + F_40(x) + F_44(x))
Eq(F_63(x), F_58(x) + F_64(x))
Eq(F_64(x), 2*F_35(x) + F_55(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_31(x) + F_67(x))
Eq(F_67(x), F_65(x))
Eq(F_68(x), F_65(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_58(x) + F_70(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_31(x) + F_70(x))
Eq(F_76(x), F_67(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_70(x) + F_77(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_75(x) + F_82(x))
Eq(F_82(x), F_18(x) + F_58(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [3, 1, 0, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [3, 1, 0, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [3, 1, 0, 2], "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": []}], "class_module": 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