0132_0231_0312_1023_2130

Counting sequence:
1, 1, 2, 6, 19, 54, 143, 362, 890, 2149, 5129, 12143, 28579, 66953, 156264, 363546, 843429, 1951918, 4507197, 10386590, 23891148, 54861093, 125780391, 287960819, 658371677, 1503372777, 3428912806, 7812198534, 17780620751, 40430061886, 91848137987, 208481378218, 472841688054, 1071605068349, 2426843103965, 5492279942799, 12421742600751, 28076635186977, 63423916125372, 143191884114250, 323111328821137, 728726790386150, 1642728207384801, 3701384153991310, 8336208135786744, 18766653296290733, 42230524791636395, 94993638846602819, 213598202061913369, 480111305105358641, 1078781382564652202, 2423135505278296678, 5441015768026002923, 12213644956852089510, 27408064033540414935, 61487054516586759498, 137900407391220890578, 309191824232930696565, 693066504497825077265, 1553137672818820173327, 3479658628151417796891, 7793962478877731099673, 17453324323160372354672, 39075022117925189220314, 87463015119711233393373, 195729633065135176825726, 437923560096753244950501, 979607687415316251300830, 2190890390602517073120388, 4898984378563843880651669, 10952434071169215519526927, 24481419168900120519708147, 54712424025544486224591413, 122253342367245202084674521, 273125885029323725328515598, 610090357645238739465057702, 1362562445024803300297619559, 3042645330939024765128866926, 6793296294804523950716445451, 15165113972563403183000642122, 33849185707785029532875520782, 75542207849094215500951874061, 168566568886120651135143289765, 376092714342483164105385262575, 838998958397693840800653104231, 1871424168624143743864549801073, 4173770808338498306392908625796, 9307471410433779059792248575946, 20753088343152514486988703688841, 46268219958712880182008053228918, 103141387091357333833289607697225, 229897535406523583596070081138318, 512374908997142203160790276833136, 1141811917299903846137635536674013, 2544224467315511889279374067154131, 5668541656471909364322246297796099, 12628251568145471445970309280877489, 28130143376719133137004387838790625, 62655377768979275732502520281786066, 139541462162143971970506527605869126, 310747393586199653552864193783240003

Generating function in Maple syntax:
-(3*x^8-6*x^7+9*x^6-15*x^5+14*x^4-16*x^3+14*x^2-6*x+1)/(x^3+2*x-1)^2/(x-1)^3

Generating function in latex syntax:
-\frac{3 x^{8}-6 x^{7}+9 x^{6}-15 x^{5}+14 x^{4}-16 x^{3}+14 x^{2}-6 x +1}{\left(x^{3}+2 x -1\right)^{2} \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-3*x**8 + 6*x**7 - 9*x**6 + 15*x**5 - 14*x**4 + 16*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(x**3 + 2*x - 1)**2)

Implicit equation for the generating function in Maple syntax:
(x^3+2*x-1)^2*(x-1)^3*F(x)+3*x^8-6*x^7+9*x^6-15*x^5+14*x^4-16*x^3+14*x^2-6*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+3 x^{8}-6 x^{7}+9 x^{6}-15 x^{5}+14 x^{4}-16 x^{3}+14 x^{2}-6 x +1 = 0

Explicit closed form in Maple syntax:
1/1416*(144*(1/165888*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/18432*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/3456*I*3^(1/2)-1/3456)^(-n)*(n+59/144)*((I*59^(1/2)-3)*(81+9*59^(1/2)*3^(1/2))^(1/3)-9*3^(1/6)*(I-1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)+8*I*2^(1/3)*3^(5/6)-8*6^(1/3))^n*((-I*59^(1/2)-3)*(81+9*59^(1/2)*3^(1/2))^(1/3)+9*3^(1/6)*(I+1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)-8*I*2^(1/3)*3^(5/6)-8*6^(1/3))^n*(18+2*59^(1/2)*3^(1/2))^(1/3*n)*(-4608*((1/16*I*59^(1/2)-3/16)*(108+12*59^(1/2)*3^(1/2))^(1/3)-3/16*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+I*3^(1/2)-1)*(-3*(81+9*59^(1/2)*3^(1/2))^(1/3)+(9+59^(1/2)*3^(1/2))^(1/3)*59^(1/2)*3^(1/6)-8*6^(1/3)))^(-n)-354*(-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)+2^(1/3)*3^(1/6)*59^(1/2))*(9+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*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)*(n^2+4*n-7/2)*(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)*(-1/1152*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/1152*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/9216*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-2*(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)*(2^(1/3)*(3^(1/6)*(n-409/1888)*59^(1/2)-3*(n+1253/288)*3^(2/3))*(9+59^(1/2)*3^(1/2))^(2/3)-8*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*(9701/22656*59^(1/2)*3^(5/6)+3^(1/3)*(n-521/128)))*(3*(81+9*59^(1/2)*3^(1/2))^(1/3)-(9+59^(1/2)*3^(1/2))^(1/3)*59^(1/2)*3^(1/6)+8*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)*(-24*(108+12*59^(1/2)*3^(1/2))^(1/3)+24*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((3*I*59^(1/2)-9)*18^(1/3)-27*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(2^(1/3)*((I*3^(2/3)+3^(1/6))*(n-409/1888)*59^(1/2)-9*(n+1253/288)*(I*3^(1/6)+1/3*3^(2/3)))*(9+59^(1/2)*3^(1/2))^(2/3)+8*2^(2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*((9701/7552*I*3^(1/3)-9701/22656*3^(5/6))*59^(1/2)+(I*3^(5/6)-3^(1/3))*(n-521/128)))*(1/6*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-3/2*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+8*I*3^(1/2)-8)^(-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)+144*(n+59/144)*(1/3456*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/384*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/72*I*3^(1/2)-1/72)^(-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))*(-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)+2^(1/3)*3^(1/6)*59^(1/2))*(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)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*((2^(1/3)*((I*3^(2/3)-3^(1/6))*(n-409/1888)*59^(1/2)-9*(n+1253/288)*(I*3^(1/6)-1/3*3^(2/3)))*(9+59^(1/2)*3^(1/2))^(2/3)+8*2^(2/3)*((9701/7552*I*3^(1/3)+9701/22656*3^(5/6))*59^(1/2)+(I*3^(5/6)+3^(1/3))*(n-521/128))*(9+59^(1/2)*3^(1/2))^(1/3))*(1/6*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-3/2*3^(5/6)*(I-1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)+8*I*3^(1/2)-8)^(-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)-144*(n+59/144)*(1/3456*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/384*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-1/72*I*3^(1/2)-1/72)^(-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:
\frac{\left(144 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{165888}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{18432}-\frac{\mathrm{I} \sqrt{3}}{3456}-\frac{1}{3456}\right)^{-n} \left(n +\frac{59}{144}\right) \left(\left(\mathrm{I} \sqrt{59}-3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{5}{6}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(\left(-\mathrm{I} \sqrt{59}-3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{1}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{5}{6}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(-4608 \left(\left(\frac{\mathrm{I} \sqrt{59}}{16}-\frac{3}{16}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{16}+\mathrm{I} \sqrt{3}-1\right) \left(-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, 3^{\frac{1}{6}}-8 \,6^{\frac{1}{3}}\right)\right)^{-n}-354 \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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \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(n^{2}+4 n -\frac{7}{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{\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(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{1152}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{1152}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{9216}\right)^{n}+\left(-2 \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(2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \left(n -\frac{409}{1888}\right) \sqrt{59}-3 \left(n +\frac{1253}{288}\right) 3^{\frac{2}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}-8 \,2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{9701 \sqrt{59}\, 3^{\frac{5}{6}}}{22656}+3^{\frac{1}{3}} \left(n -\frac{521}{128}\right)\right)\right) \left(3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, 3^{\frac{1}{6}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{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(-24 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+24 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) 18^{\frac{1}{3}}-27 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\left(2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \left(n -\frac{409}{1888}\right) \sqrt{59}-9 \left(n +\frac{1253}{288}\right) \left(\mathrm{I} \,3^{\frac{1}{6}}+\frac{3^{\frac{2}{3}}}{3}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+8 \,2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\frac{9701 \,\mathrm{I} \,3^{\frac{1}{3}}}{7552}-\frac{9701 \,3^{\frac{5}{6}}}{22656}\right) \sqrt{59}+\left(\mathrm{I} \,3^{\frac{5}{6}}-3^{\frac{1}{3}}\right) \left(n -\frac{521}{128}\right)\right)\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}+8 \,\mathrm{I} \sqrt{3}-8\right)^{-n} \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}+144 \left(n +\frac{59}{144}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \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}\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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\left(2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \left(n -\frac{409}{1888}\right) \sqrt{59}-9 \left(n +\frac{1253}{288}\right) \left(\mathrm{I} \,3^{\frac{1}{6}}-\frac{3^{\frac{2}{3}}}{3}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+8 \,2^{\frac{2}{3}} \left(\left(\frac{9701 \,\mathrm{I} \,3^{\frac{1}{3}}}{7552}+\frac{9701 \,3^{\frac{5}{6}}}{22656}\right) \sqrt{59}+\left(\mathrm{I} \,3^{\frac{5}{6}}+3^{\frac{1}{3}}\right) \left(n -\frac{521}{128}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}+8 \,\mathrm{I} \sqrt{3}-8\right)^{-n} \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}-144 \left(n +\frac{59}{144}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \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}\right)\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{1416}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 143
a(7) = 362
a(8) = 890
a(n) = -4*a(n+2)+2*a(n+3)-4*a(n+4)+4*a(n+5)-a(n+6)-n*(n+6), n >= 9

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) = 143
a \! \left(7\right) = 362
a \! \left(8\right) = 890
a \! \left(n \right) = -4 a \! \left(n +2\right)+2 a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-a \! \left(n +6\right)-n \left(n +6\right), \quad n \geq 9

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18587/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[21,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[4,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[70,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[37,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[23,x]+F[28,x]+F[34,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]+F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[20,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[14,x]+F[36,x]
F[36,x] = F[34,x]
F[37,x] = F[38,x]+F[61,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[14,x]+F[42,x]
F[42,x] = F[28,x]
F[43,x] = F[38,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[57,x]
F[47,x] = F[48,x]+F[55,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[4,x]+F[52,x]
F[52,x] = F[28,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = F[45,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[47,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[48,x]+F[58,x]
F[61,x] = 2*F[23,x]+F[45,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[36,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[36,x]
F[69,x] = F[66,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[90,x]
F[72,x] = F[2,x]+F[73,x]
F[73,x] = F[23,x]+F[74,x]+F[89,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[4,x]+F[77,x]
F[77,x] = F[23,x]+F[28,x]+F[78,x]
F[78,x] = F[14,x]*F[4,x]
F[79,x] = F[48,x]+F[80,x]
F[80,x] = 2*F[23,x]+F[45,x]+F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[78,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[78,x]
F[88,x] = F[85,x]
F[89,x] = F[4,x]*F[72,x]
F[90,x] = F[91,x]+F[94,x]
F[91,x] = F[23,x]+F[24,x]+F[92,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[2,x]+F[91,x]
F[94,x] = F[59,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{70}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \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_{31}\! \left(x \right)+F_{4}\! \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_{20}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{61}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{28}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{57}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{55}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \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_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{45}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \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_{48}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{36}\! \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_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{36}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{90}\! \left(x \right)
F_{72}\! \left(x \right) = F_{2}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{23}\! \left(x \right)+F_{74}\! \left(x \right)+F_{89}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{79}\! \left(x \right) = F_{48}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{45}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{78}\! \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_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{78}\! \left(x \right)
F_{88}\! \left(x \right) = F_{85}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{94}\! \left(x \right)
F_{91}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{2}\! \left(x \right)+F_{91}\! \left(x \right)
F_{94}\! \left(x \right) = F_{59}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_21(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_4(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_70(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_37(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_23(x) + F_28(x) + F_34(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_31(x) + F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_20(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_14(x) + F_36(x))
Eq(F_36(x), F_34(x))
Eq(F_37(x), F_38(x) + F_61(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_14(x) + F_42(x))
Eq(F_42(x), F_28(x))
Eq(F_43(x), F_38(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_57(x))
Eq(F_47(x), F_48(x) + F_55(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_4(x) + F_52(x))
Eq(F_52(x), F_28(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), F_45(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_47(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_48(x) + F_58(x))
Eq(F_61(x), 2*F_23(x) + F_45(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_36(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_36(x))
Eq(F_69(x), F_66(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_90(x))
Eq(F_72(x), F_2(x) + F_73(x))
Eq(F_73(x), F_23(x) + F_74(x) + F_89(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_4(x) + F_77(x))
Eq(F_77(x), F_23(x) + F_28(x) + F_78(x))
Eq(F_78(x), F_14(x)*F_4(x))
Eq(F_79(x), F_48(x) + F_80(x))
Eq(F_80(x), 2*F_23(x) + F_45(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_78(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_78(x))
Eq(F_88(x), F_85(x))
Eq(F_89(x), F_4(x)*F_72(x))
Eq(F_90(x), F_91(x) + F_94(x))
Eq(F_91(x), F_23(x) + F_24(x) + F_92(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_2(x) + F_91(x))
Eq(F_94(x), F_59(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:
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