Av(1234, 1342, 1423, 2314, 3241)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{8}-5 x^{7}+10 x^{6}-12 x^{5}+16 x^{4}-19 x^{3}+15 x^{2}-6 x +1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{5}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 54, 143, 360, 871, 2046, 4706, 10667, 23940, 53378, 118514, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-x^{8}+5 x^{7}-10 x^{6}+12 x^{5}-16 x^{4}+19 x^{3}-15 x^{2}+6 x -1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(7\right) = 360\)
\(\displaystyle a \! \left(8\right) = 871\)
\(\displaystyle a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right)-\frac{\left(n +1\right) \left(n^{3}+5 n^{2}+18 n +48\right)}{24}, \quad n \geq 9\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}\frac{473 \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 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\frac{1888}{1419} n^{4}+\frac{7552}{1419} n^{3}+\frac{37760}{1419} n^{2}+\frac{139712}{1419} n +\frac{42480}{473}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}-\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}+\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\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}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\left(-\frac{3^{\frac{5}{6}}}{3}+\mathrm{I} \,3^{\frac{1}{3}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}+\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{2 \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} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{n} \left(\frac{1}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n} \left(\left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{413 \,3^{\frac{1}{3}}}{473}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{961 \,2^{\frac{1}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{946}\right) \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-13824 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-13824 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-1728 \,\mathrm{I} \sqrt{59}-5184\right) 18^{\frac{1}{3}}+15552 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+1728 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{21240 \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3456 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-18 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+96 \,\mathrm{I} \sqrt{3}-96\right)^{-n}}{473}\right)}{3}\right)}{30208} & n <0 \\ 1 & n =0 \\ \frac{473 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(-\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(\frac{1888}{1419} n^{4}+\frac{7552}{1419} n^{3}+\frac{37760}{1419} n^{2}+\frac{139712}{1419} n +\frac{42480}{473}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}-\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}+\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}+\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\left(\left(-\frac{3^{\frac{5}{6}}}{3}+\mathrm{I} \,3^{\frac{1}{3}}\right) \sqrt{59}-\frac{413 \,\mathrm{I} \,3^{\frac{5}{6}}}{1419}+\frac{413 \,3^{\frac{1}{3}}}{1419}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{14160}{473}+\frac{961 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{20001 \,\mathrm{I} \,3^{\frac{1}{6}}}{961}-\frac{6667 \,3^{\frac{2}{3}}}{961}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2838}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}-\frac{961 \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} \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(-\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(\left(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(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}} \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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-13824 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-13824 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-1728 \,\mathrm{I} \sqrt{59}-5184\right) 18^{\frac{1}{3}}+15552 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+1728 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{1419}+\frac{14160 \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\frac{1}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{165888}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{18432}+\frac{\mathrm{I} \sqrt{3}}{3456}-\frac{1}{3456}\right)^{-n} \left(\frac{\left(-\mathrm{I} \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{96}+\frac{3 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}}{12}+\frac{6^{\frac{1}{3}}}{12}\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}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{473}+\frac{2 \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{413 \,3^{\frac{1}{3}}}{473}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{6667 \,2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{946}\right) \left(-\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\left(\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}}\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} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{1728}+\frac{3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{192}-\frac{\mathrm{I} \sqrt{3}}{36}-\frac{1}{36}\right)^{-n}}{3}\right)}{30208} & 0<n \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 155 rules.

Found on January 18, 2022.

Finding the specification took 5 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 155 equations to clipboard:
\(\begin{align*} 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_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{96}\! \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_{38}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\ 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_{30}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \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_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \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_{48}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{7}\! \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_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{43}\! \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_{43}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{54}\! \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_{67}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{62}\! \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_{27}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{91}\! \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_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{153}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \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_{128}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{2}\! \left(x \right)\\ \end{align*}\)