0123_0231_0312_1203_3102
Counting sequence:
1, 1, 2, 6, 19, 53, 134, 320, 739, 1673, 3744, 8324, 18439, 40763, 90016, 198664, 438313, 966895, 2132740, 4704114, 10375481, 22884099, 50472750, 111321462, 245527549, 541528421, 1194378926, 2634286074, 5810101295, 12814582297, 28263451506, 62337005204, 137488593663, 303240639853, 668818285996, 1475125166808, 3253490974691, 7175800236671, 15826725641516, 34906942259164, 76989684756517, 169806095156147, 374519132573136, 826027949904550, 1821861994967093, 4018243122509255, 8862514194925082, 19546890384819370, 43112023892150201, 95086561979227785, 209720014343277338, 462552052578707374, 1020190667136645131, 2250101348616570301, 4962754749811850782, 10945700166760349608, 24141501682137272539, 53245758114086398993, 117437216394933150840, 259015934472003577580, 571277627058093557631, 1259992470511120269699, 2779000875494244120696, 6129279378046581802864, 13518551226604283879393, 29816103328702811883575, 65761486035452205574236, 145041523297508695032218, 319899149923720201952497, 705559785882892609483851, 1556161095063293914004678, 3432221340050308029966750, 7570002465983508669422389, 16696166027030311252854637, 36824553394110930535681350, 81219109254205369740790562, 179134384535441050734441383, 395093322464993032004569889, 871405754184191433749936266, 1921945892903823918234319996, 4238985108272640868473216119, 9349375970729473170696374901, 20620697834362770259627076356, 45480380776998181387727375552, 100310137524725835946151132891, 221240972883814442151929349191, 487962326544627065691586081156, 1076234790613979967329323302596, 2373710554111774376810575961949, 5235383434768175819312738012795, 11547001660150331605954799336104, 25467713874412437588720174642254, 56170811183593050996753087305581, 123888624027336433599460973955727, 273244961929085304787642122562354, 602660735041763660572037332439122, 1329210094110863754743535638842993, 2931665150150812814274713400257553, 6465991035343389289121464132963634, 14261192164797642332986463904779862, 31454049479746097480247641209827075
Generating function in Maple syntax:
-(2*x^5-4*x^4+3*x^3-6*x^2+4*x-1)/(x^3+2*x-1)/(-1+x)^3
Generating function in latex syntax:
-\frac{2 x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1}{\left(x^{3}+2 x -1\right) \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(-2*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)*(-1+x)^3*F(x)+2*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(-1+x \right)^{3} F \! \left(x \right)+2 x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
37/2832*(108+12*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*(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)*(1/27648*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/3072*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/576*I*3^(1/2)-1/576)^(-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)*(2^(2/3)*(59^(1/2)*3^(5/6)-177/37*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)+413/74*2^(1/3)*3^(2/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)+48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((6*I*59^(1/2)-18)*18^(1/3)-54*I*3^(1/6)*2^(1/3)+6*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/576*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/64*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(1/2)-1/12)^(-n)*(((I*3^(1/3)-1/3*3^(5/6))*59^(1/2)-59/37*I*3^(5/6)+59/37*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+13/74*((I*3^(2/3)+3^(1/6))*59^(1/2)-413/13*I*3^(1/6)-413/39*3^(2/3))*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-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)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-1416/37*((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)*((n^2+3*n)*(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-n)+13/1416*2^(1/3)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(2/3)*(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)*(1/27648*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/3072*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/576*I*3^(1/2)-1/576)^(-n)*59^(1/2))*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)-48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-6*I*59^(1/2)-18)*18^(1/3)+54*I*3^(1/6)*2^(1/3)+6*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n-(1/576*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/64*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(1/2)-1/12)^(-n)*(((I*3^(1/3)+1/3*3^(5/6))*59^(1/2)-59/37*I*3^(5/6)-59/37*3^(1/3))*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)+13/74*((I*3^(2/3)-3^(1/6))*59^(1/2)-413/13*I*3^(1/6)+413/39*3^(2/3))*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3))*(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-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)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+944/37*(1/576*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/64*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/12*I*3^(1/2)-1/12)^(-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)*((-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*3^(1/6)*2^(1/3)+59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)-48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-6*I*59^(1/2)-18)*18^(1/3)+54*I*3^(1/6)*2^(1/3)+6*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)+(-59^(1/2)*3^(1/6)*2^(1/3)+3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)+(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)-48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-6*I*59^(1/2)-18)*18^(1/3)+54*I*3^(1/6)*2^(1/3)+6*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)+48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((6*I*59^(1/2)-18)*18^(1/3)-54*I*3^(1/6)*2^(1/3)+6*59^(1/2)*3^(1/6)*2^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^n))
Explicit closed form in latex syntax:
\frac{37 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(\frac{2 \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} \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{27648}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3072}-\frac{\mathrm{I} \sqrt{3}}{576}-\frac{1}{576}\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(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{177 \,3^{\frac{1}{3}}}{37}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{413 \,2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{74}\right) \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}-54 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+6 \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}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(\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}}}{37}+\frac{59 \,3^{\frac{1}{3}}}{37}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{13 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{1}{6}}}{13}-\frac{413 \,3^{\frac{2}{3}}}{39}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{74}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}\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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\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{1416 \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} \left(\left(n^{2}+3 n \right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}+\frac{13 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \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{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{27648}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3072}-\frac{\mathrm{I} \sqrt{3}}{576}-\frac{1}{576}\right)^{-n} \sqrt{59}}{1416}\right) \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+6 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{37}-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(\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}}}{37}-\frac{59 \,3^{\frac{1}{3}}}{37}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{13 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{1}{6}}}{13}+\frac{413 \,3^{\frac{2}{3}}}{39}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{74}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}\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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\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{944 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \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} \left(\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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+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(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(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+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(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}-54 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+6 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}\right)}{37}\right)}{2832}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(n) = -n^2-2*a(n+2)+a(n+3)-4*n-1, n >= 6
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) = 53
a \! \left(n \right) = -n^{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 n -1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17020/
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[68,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[50,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[25,x]+F[49,x]
F[49,x] = F[43,x]
F[50,x] = F[34,x]+F[51,x]
F[51,x] = 2*F[35,x]+F[52,x]+F[62,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[31,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[31,x]+F[55,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[31,x]+F[59,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[33,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[31,x]+F[65,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[30,x]+F[70,x]
F[70,x] = F[71,x]+F[84,x]
F[71,x] = F[35,x]+F[72,x]+F[82,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[78,x]
F[74,x] = F[11,x]+F[75,x]
F[75,x] = F[35,x]+F[44,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[42,x]
F[78,x] = F[31,x]+F[79,x]
F[79,x] = 2*F[35,x]+F[56,x]+F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[54,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[18,x]+F[71,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[31,x]+F[84,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_{68}\! \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_{50}\! \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_{25}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)
F_{50}\! \left(x \right) = F_{34}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{31}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{33}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{31}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{30}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{84}\! \left(x \right)
F_{71}\! \left(x \right) = F_{35}\! \left(x \right)+F_{72}\! \left(x \right)+F_{82}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{11}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{42}\! \left(x \right)
F_{78}\! \left(x \right) = F_{31}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{56}\! \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_{54}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{18}\! \left(x \right)+F_{71}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{31}\! \left(x \right)+F_{84}\! \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_68(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_50(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_25(x) + F_49(x))
Eq(F_49(x), F_43(x))
Eq(F_50(x), F_34(x) + F_51(x))
Eq(F_51(x), 2*F_35(x) + F_52(x) + F_62(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_31(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_31(x) + F_55(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_31(x) + F_59(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_33(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_31(x) + F_65(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_30(x) + F_70(x))
Eq(F_70(x), F_71(x) + F_84(x))
Eq(F_71(x), F_35(x) + F_72(x) + F_82(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_78(x))
Eq(F_74(x), F_11(x) + F_75(x))
Eq(F_75(x), F_35(x) + F_44(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_42(x))
Eq(F_78(x), F_31(x) + F_79(x))
Eq(F_79(x), 2*F_35(x) + F_56(x) + F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_54(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_18(x) + F_71(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_31(x) + F_84(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, 2, 0, 3], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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