Av(1432, 2413, 3142, 3214, 3412)
Generating Function
\(\displaystyle \frac{x^{6}-x^{5}+4 x^{4}-3 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 140, 341, 798, 1822, 4099, 9142, 20288, 44897, 99202, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{6}+x^{5}-4 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 140\)
\(\displaystyle a \! \left(n \right) = -2 n^{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-2 n -2, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 140\)
\(\displaystyle a \! \left(n \right) = -2 n^{2}-2 a \! \left(n +2\right)+a \! \left(n +3\right)-2 n -2, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}\frac{23 \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(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(-\frac{1888 \left(n^{2}+\frac{3}{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{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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}}{23}-\frac{37 \sqrt{59}\, 3^{n +\frac{1}{6}} 2^{4 n +\frac{1}{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(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{69}-\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(2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}-\frac{59 \,3^{\frac{1}{3}}}{69}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{944}{23}+\frac{37 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}+\frac{295 \,3^{\frac{2}{3}}}{37}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right)\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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-\frac{\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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(\frac{944 \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(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-4 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n} 331776^{n}}{23}+\frac{2 \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(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{295 \,3^{n +\frac{2}{3}} 2^{4 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{46}+2^{4 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\sqrt{59}\, 3^{n +\frac{5}{6}}-\frac{59 \,3^{n +\frac{1}{3}}}{23}\right)\right) \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}-31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{3}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{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(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}+\frac{59 \,3^{\frac{1}{3}}}{69}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{944}{23}+\frac{37 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}-\frac{295 \,3^{\frac{2}{3}}}{37}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\right)\right)\right)}{1888} & n <0 \\ 1 & n =0 \\ \frac{23 \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 n^{2}}{23}+\frac{2832}{23}\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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}-\frac{59 \,3^{\frac{1}{3}}}{69}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{944}{23}+\frac{37 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}+\frac{295 \,3^{\frac{2}{3}}}{37}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(\frac{\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(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}+\frac{59 \,3^{\frac{1}{3}}}{69}-\frac{59 \,\mathrm{I} \,3^{\frac{5}{6}}}{69}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{944}{23}+\frac{37 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{885 \,\mathrm{I} \,3^{\frac{1}{6}}}{37}-\frac{295 \,3^{\frac{2}{3}}}{37}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{138}\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} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{2 \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(3^{\frac{1}{6}} \sqrt{59}\, \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(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{59 \,3^{\frac{1}{3}}}{23}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{295 \,2^{\frac{1}{3}} 3^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{46}\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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-4 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-4 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n}-\frac{37 \left(\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(-288 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-288 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-36 \,\mathrm{I} \sqrt{59}-108\right) 18^{\frac{1}{3}}+324 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+36 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(3^{\frac{1}{6}} \sqrt{59}\, \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(48 \,\mathrm{I} \sqrt{59}-144\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-432 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+384 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-384 \,6^{\frac{1}{3}}\right)^{n}-\frac{2832 \left(-27648 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-27648 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-3456 \,\mathrm{I} \sqrt{59}-10368\right) 18^{\frac{1}{3}}+31104 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3456 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(2 \,3^{\frac{1}{6}} \sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-6 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-16 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}}}{37}\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}}{46}\right)}{3}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{1888} & 0<n \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 73 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 73 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_{24}\! \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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \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_{41}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{46}\! \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_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \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_{71}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{69}\! \left(x \right)\\
\end{align*}\)