0123_0132_0231_1302_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 141, 350, 842, 1985, 4617, 10639, 24349, 55437, 125696, 284034, 640005, 1438606, 3226931, 7225106, 16151148, 36053721, 80381351, 179011251, 398270799, 885311853, 1966412494, 4364650566, 9681722223, 21464075070, 47561236593, 105341066782, 233219950950, 516148622665, 1141933279965, 2525676446063, 5584681384601, 12345655788197, 27285707499892, 60293535338594, 133207604373105, 294250672060806, 649894391088839, 1435195409807234, 3169039538187320, 6996769560486673, 15446326716826027, 34096877343929283, 75260892992524027, 166108850190230581, 366596052701102322, 809015948348151254, 1785266646793377035, 3939381146101543014, 8692281840178609205, 19178837526405337918, 42315070597421702370, 93358451832040979505, 205967798784525226513, 454394783354548012975, 1002436248917288718965, 2211416757371406089757, 4878381219601967041352, 10761504531130436497570, 23739037505650706474813, 52365679602940234768862, 115511310481084615589051, 254798551955967356758354, 562036570491169786494308, 1239732025416013864992937, 2734557750693174439572495, 6031742367687877371293491, 13304397352412486018885895, 29345713638759581299953757, 64727892011689909616417398, 142769626108758044542150854, 314903855322207148965113095, 694573381587967164708356078, 1531991947148418288282285145, 3379038865346495554176524734, 7452953343735861930355085246, 16438503097529949463579811961, 36257253986226009110510858149, 79969879167827109409726217295, 176383097136462626799731074737, 389033119665708179743370660597, 858055461312357302963262840956, 1892532705387404900859847358018, 4174175901692973317730246575721, 9206562007203214610958118386838, 20305966204803655467844808916751, 44786727281319926943557313975426, 98781254509882353878347645466096, 217870951104646933985089898101665, 480533581250770936434836706680019, 1059858320531738509790220251824387, 2337607399208752519649928787742579, 5155787993749533107903491054144997, 11371513536193318989934795904069466, 25080792927920419028194707683787030, 55317690762240428221643280597525347
Generating function in Maple syntax:
-(4*x^6-9*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{4 x^{6}-9 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:
(-4*x**6 + 9*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)+4*x^6-9*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)+4 x^{6}-9 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
197/45312*(32096/197*(1/27*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/3*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-16/9*I*3^(1/2)-16/9)^(-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*3^(5/6)*2^(1/3)+8*6^(1/3))/(59^(1/2)*3^(1/6)*(9+59^(1/2)*3^(1/2))^(1/3)-8*6^(1/3)-3*(81+9*59^(1/2)*3^(1/2))^(1/3)))^n*(18+2*59^(1/2)*3^(1/2))^(1/3*n)*((-64*I*59^(1/2)-192)*(81+9*59^(1/2)*3^(1/2))^(1/3)+576*3^(1/6)*(I+1/9*59^(1/2))*(9+59^(1/2)*3^(1/2))^(1/3)-512*I*3^(5/6)*2^(1/3)-512*6^(1/3))^n+1916/591*(-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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(2^(2/3)*(59^(1/2)*3^(5/6)-1947/479*3^(1/3))*(9+59^(1/2)*3^(1/2))^(1/3)-591/958*2^(1/3)*(59^(1/2)*3^(1/6)-5605/591*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)+(2*2^(1/3)*3^(1/6)*59^(1/2)-6*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3)+(12*I-4*3^(1/2))*59^(1/2)-36*I*3^(1/2)+36)^(-n)-11328/197*(1/9*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-16/3*I*3^(1/2)-16/3)^(-n)*((n^2+4*n-7/2)*(-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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-17/6*576^n)*(-16/3*(108+12*59^(1/2)*3^(1/2))^(1/3)+16/3*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/3*((2*I*59^(1/2)-6)*18^(1/3)-18*I*2^(1/3)*3^(1/6)+2*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+32096/197*576^n*(1/9*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-16/3*I*3^(1/2)-16/3)^(-n)*(-16/3*(108+12*59^(1/2)*3^(1/2))^(1/3)-16/3*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/3*((-2*I*59^(1/2)-6)*18^(1/3)+18*I*2^(1/3)*3^(1/6)+2*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-90624/197*(1/128*(-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9/128*(I+1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)-3/8*I*3^(1/2)-3/8)^(-n)*(-3/4*(108+12*59^(1/2)*3^(1/2))^(1/3)+3/4*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/32*((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*(-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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+((3^(2*n+2/3)*(I*59^(1/2)-5605/591)*(9+59^(1/2)*3^(1/2))^(2/3)*2^(6*n+1/3)+958/197*2^(6*n+2/3)*(I*59^(1/2)+649/479)*(9+59^(1/2)*3^(1/2))^(1/3)*3^(2*n+1/3))*(-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)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-3^(2*n+2/3)*(I*59^(1/2)+5605/591)*(9+59^(1/2)*3^(1/2))^(2/3)*2^(6*n+1/3)-958/197*2^(6*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*(I*59^(1/2)-649/479)*3^(2*n+1/3))*(-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)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(-1298/197*2^(2/3)*(479/1947*59^(1/2)+I)*3^(5/6)*(9+59^(1/2)*3^(1/2))^(1/3)-5605/197*3^(1/6)*2^(1/3)*(-197/5605*59^(1/2)+I)*(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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+1298/197*((I-479/1947*59^(1/2))*2^(2/3)*3^(5/6)*(9+59^(1/2)*3^(1/2))^(1/3)+95/22*3^(1/6)*2^(1/3)*(9+59^(1/2)*3^(1/2))^(2/3)*(I+197/5605*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*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(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))*(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{197 \left(\frac{32096 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{27}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \sqrt{3}}{9}-\frac{16}{9}\right)^{-n} \left(\frac{\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} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(\left(-64 \,\mathrm{I} \sqrt{59}-192\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+576 \,3^{\frac{1}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-512 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-512 \,6^{\frac{1}{3}}\right)^{n}}{197}+\frac{1916 \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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+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(-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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+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(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{1947 \,3^{\frac{1}{3}}}{479}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{591 \,2^{\frac{1}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}}-\frac{5605 \,3^{\frac{2}{3}}}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{958}\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(2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}-6 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(12 \,\mathrm{I}-4 \sqrt{3}\right) \sqrt{59}-36 \,\mathrm{I} \sqrt{3}+36\right)^{-n}}{591}-\frac{11328 \left(\frac{\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}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(\left(n^{2}+4 n -\frac{7}{2}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}-\frac{17 \,576^{n}}{6}\right) \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}-18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{197}+\frac{32096 \,576^{n} \left(\frac{\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}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}+18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{197}-\frac{90624 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}+\frac{9 \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}-\frac{3 \,\mathrm{I} \sqrt{3}}{8}-\frac{3}{8}\right)^{-n} \left(-\frac{3 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}+\frac{3 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{4}+\frac{\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}}}{32}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{197}+\left(\left(3^{2 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}-\frac{5605}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{6 n +\frac{1}{3}}+\frac{958 \,2^{6 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{649}{479}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}}}{197}\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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-3^{2 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{5605}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{6 n +\frac{1}{3}}-\frac{958 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\mathrm{I} \sqrt{59}-\frac{649}{479}\right) 3^{2 n +\frac{1}{3}}}{197}\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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-\frac{1298 \,2^{\frac{2}{3}} \left(\frac{479 \sqrt{59}}{1947}+\mathrm{I}\right) 3^{\frac{5}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{197}-\frac{5605 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(-\frac{197 \sqrt{59}}{5605}+\mathrm{I}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{197}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{1298 \left(\left(\mathrm{I}-\frac{479 \sqrt{59}}{1947}\right) 2^{\frac{2}{3}} 3^{\frac{5}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{95 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I}+\frac{197 \sqrt{59}}{5605}\right)}{22}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{197}\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(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \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}}{45312}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 141
a(n+4) = -1/2*n^2-2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)-3/2*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) = 54
a \! \left(6\right) = 141
a \! \left(n +4\right) = -\frac{n^{2}}{2}-2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-\frac{3 n}{2}+4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16127/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]
F[15,x] = F[4,x]
F[16,x] = F[12,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[120,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[13,x]+F[19,x]+F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[7,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[12,x]
F[36,x] = F[32,x]
F[37,x] = F[100,x]+F[38,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[42,x]+F[7,x]
F[42,x] = F[13,x]
F[43,x] = F[38,x]+F[44,x]
F[44,x] = 2*F[19,x]+F[45,x]+F[67,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[52,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[16,x]+F[51,x]
F[51,x] = F[32,x]
F[52,x] = F[53,x]+F[58,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[42,x]
F[57,x] = F[53,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[48,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[48,x]
F[66,x] = F[63,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[95,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[73,x]+F[74,x]
F[73,x] = F[12,x]+F[4,x]
F[74,x] = F[70,x]+F[75,x]
F[75,x] = 2*F[19,x]+F[67,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[80,x]
F[78,x] = F[12,x]+F[79,x]
F[79,x] = F[49,x]
F[80,x] = F[81,x]+F[86,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[12,x]
F[85,x] = F[81,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[79,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[93,x]+F[94,x]
F[93,x] = F[79,x]
F[94,x] = F[91,x]
F[95,x] = F[96,x]
F[96,x] = F[4,x]*F[97,x]
F[97,x] = F[98,x]+F[99,x]
F[98,x] = F[70,x]
F[99,x] = F[96,x]
F[100,x] = 2*F[19,x]+F[101,x]+F[67,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[105,x]
F[103,x] = F[104,x]+F[23,x]
F[104,x] = F[49,x]
F[105,x] = F[106,x]+F[111,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[23,x]
F[110,x] = F[106,x]
F[111,x] = F[112,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[114,x]+F[115,x]
F[114,x] = F[104,x]
F[115,x] = F[116,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]+F[119,x]
F[118,x] = F[104,x]
F[119,x] = F[116,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[122,x]+F[151,x]
F[122,x] = F[123,x]+F[2,x]
F[123,x] = F[124,x]+F[150,x]+F[19,x]
F[124,x] = F[125,x]*F[4,x]
F[125,x] = F[126,x]+F[129,x]
F[126,x] = F[127,x]+F[4,x]
F[127,x] = F[128,x]+F[13,x]+F[19,x]
F[128,x] = F[4,x]*F[7,x]
F[129,x] = F[130,x]+F[70,x]
F[130,x] = 2*F[19,x]+F[131,x]+F[67,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[133,x]+F[135,x]
F[133,x] = F[127,x]+F[134,x]
F[134,x] = F[49,x]
F[135,x] = F[136,x]+F[141,x]
F[136,x] = F[137,x]
F[137,x] = F[138,x]*F[4,x]
F[138,x] = F[139,x]+F[140,x]
F[139,x] = F[127,x]
F[140,x] = F[136,x]
F[141,x] = F[142,x]
F[142,x] = F[143,x]*F[4,x]
F[143,x] = F[144,x]+F[145,x]
F[144,x] = F[134,x]
F[145,x] = F[146,x]
F[146,x] = F[147,x]*F[4,x]
F[147,x] = F[148,x]+F[149,x]
F[148,x] = F[134,x]
F[149,x] = F[146,x]
F[150,x] = F[2,x]*F[4,x]
F[151,x] = F[152,x]+F[158,x]
F[152,x] = F[153,x]
F[153,x] = F[154,x]*F[4,x]
F[154,x] = F[155,x]+F[157,x]
F[155,x] = F[156,x]+F[2,x]
F[156,x] = F[150,x]
F[157,x] = F[152,x]
F[158,x] = F[96,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{120}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{100}\! \left(x \right)+F_{38}\! \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_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{67}\! \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_{52}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \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_{16}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{32}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{42}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \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_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{48}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{48}\! \left(x \right)
F_{66}\! \left(x \right) = F_{63}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{95}\! \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_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{74}\! \left(x \right) = F_{70}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{67}\! \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_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{49}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{86}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{79}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{94}\! \left(x \right)
F_{93}\! \left(x \right) = F_{79}\! \left(x \right)
F_{94}\! \left(x \right) = F_{91}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)+F_{99}\! \left(x \right)
F_{98}\! \left(x \right) = F_{70}\! \left(x \right)
F_{99}\! \left(x \right) = F_{96}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{101}\! \left(x \right)+F_{67}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{105}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{23}\! \left(x \right)
F_{104}\! \left(x \right) = F_{49}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{111}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{23}\! \left(x \right)
F_{110}\! \left(x \right) = F_{106}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{115}\! \left(x \right)
F_{114}\! \left(x \right) = F_{104}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{119}\! \left(x \right)
F_{118}\! \left(x \right) = F_{104}\! \left(x \right)
F_{119}\! \left(x \right) = F_{116}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{151}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{2}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{150}\! \left(x \right)+F_{19}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{4}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{129}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{128}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{70}\! \left(x \right)
F_{130}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{131}\! \left(x \right)+F_{67}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{135}\! \left(x \right)
F_{133}\! \left(x \right) = F_{127}\! \left(x \right)+F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{49}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{141}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{4}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{140}\! \left(x \right)
F_{139}\! \left(x \right) = F_{127}\! \left(x \right)
F_{140}\! \left(x \right) = F_{136}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right) F_{4}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)+F_{145}\! \left(x \right)
F_{144}\! \left(x \right) = F_{134}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right) F_{4}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)+F_{149}\! \left(x \right)
F_{148}\! \left(x \right) = F_{134}\! \left(x \right)
F_{149}\! \left(x \right) = F_{146}\! \left(x \right)
F_{150}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right)+F_{158}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right) F_{4}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right)+F_{157}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right)+F_{2}\! \left(x \right)
F_{156}\! \left(x \right) = F_{150}\! \left(x \right)
F_{157}\! \left(x \right) = F_{152}\! \left(x \right)
F_{158}\! \left(x \right) = F_{96}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_12(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_120(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_13(x) + F_19(x) + F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_32(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_31(x))
Eq(F_30(x), F_7(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_12(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_100(x) + F_38(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_42(x) + F_7(x))
Eq(F_42(x), F_13(x))
Eq(F_43(x), F_38(x) + F_44(x))
Eq(F_44(x), 2*F_19(x) + F_45(x) + F_67(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_52(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_16(x) + F_51(x))
Eq(F_51(x), F_32(x))
Eq(F_52(x), F_53(x) + F_58(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_42(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_48(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_48(x))
Eq(F_66(x), F_63(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_95(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_73(x) + F_74(x))
Eq(F_73(x), F_12(x) + F_4(x))
Eq(F_74(x), F_70(x) + F_75(x))
Eq(F_75(x), 2*F_19(x) + F_67(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_80(x))
Eq(F_78(x), F_12(x) + F_79(x))
Eq(F_79(x), F_49(x))
Eq(F_80(x), F_81(x) + F_86(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_12(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_79(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_93(x) + F_94(x))
Eq(F_93(x), F_79(x))
Eq(F_94(x), F_91(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), F_4(x)*F_97(x))
Eq(F_97(x), F_98(x) + F_99(x))
Eq(F_98(x), F_70(x))
Eq(F_99(x), F_96(x))
Eq(F_100(x), F_101(x) + 2*F_19(x) + F_67(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_105(x))
Eq(F_103(x), F_104(x) + F_23(x))
Eq(F_104(x), F_49(x))
Eq(F_105(x), F_106(x) + F_111(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_23(x))
Eq(F_110(x), F_106(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_114(x) + F_115(x))
Eq(F_114(x), F_104(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x) + F_119(x))
Eq(F_118(x), F_104(x))
Eq(F_119(x), F_116(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_122(x) + F_151(x))
Eq(F_122(x), F_123(x) + F_2(x))
Eq(F_123(x), F_124(x) + F_150(x) + F_19(x))
Eq(F_124(x), F_125(x)*F_4(x))
Eq(F_125(x), F_126(x) + F_129(x))
Eq(F_126(x), F_127(x) + F_4(x))
Eq(F_127(x), F_128(x) + F_13(x) + F_19(x))
Eq(F_128(x), F_4(x)*F_7(x))
Eq(F_129(x), F_130(x) + F_70(x))
Eq(F_130(x), F_131(x) + 2*F_19(x) + F_67(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_133(x) + F_135(x))
Eq(F_133(x), F_127(x) + F_134(x))
Eq(F_134(x), F_49(x))
Eq(F_135(x), F_136(x) + F_141(x))
Eq(F_136(x), F_137(x))
Eq(F_137(x), F_138(x)*F_4(x))
Eq(F_138(x), F_139(x) + F_140(x))
Eq(F_139(x), F_127(x))
Eq(F_140(x), F_136(x))
Eq(F_141(x), F_142(x))
Eq(F_142(x), F_143(x)*F_4(x))
Eq(F_143(x), F_144(x) + F_145(x))
Eq(F_144(x), F_134(x))
Eq(F_145(x), F_146(x))
Eq(F_146(x), F_147(x)*F_4(x))
Eq(F_147(x), F_148(x) + F_149(x))
Eq(F_148(x), F_134(x))
Eq(F_149(x), F_146(x))
Eq(F_150(x), F_2(x)*F_4(x))
Eq(F_151(x), F_152(x) + F_158(x))
Eq(F_152(x), F_153(x))
Eq(F_153(x), F_154(x)*F_4(x))
Eq(F_154(x), F_155(x) + F_157(x))
Eq(F_155(x), F_156(x) + F_2(x))
Eq(F_156(x), F_150(x))
Eq(F_157(x), F_152(x))
Eq(F_158(x), F_96(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [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": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 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