Av(1234, 1324, 1432, 2143, 2413)
View Raw Data
Generating Function
\(\displaystyle -\frac{2 x -1}{x^{5}-2 x^{4}-x^{3}+x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 152, 431, 1227, 3491, 9927, 28227, 80268, 228259, 649099, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-2 x^{4}-x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{139 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{56}{33}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{56 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}+\frac{127}{33}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\frac{56 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}+\frac{127}{33}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{127 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}-\frac{42}{11}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{56 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}+\frac{127}{33}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{127 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}-\frac{42}{11}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{127 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{33}-\frac{42}{11}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{42 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{11}-\frac{63}{11}\right) \left(\left(\left(\left(-1+\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{218}{139}-\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{165}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{606}{139}-\frac{288 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1118}{139}+\frac{544 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{51 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{684}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{218}{139}-\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{1118}{139}+\frac{544 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{117 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{2022}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{7}{139}-\frac{686 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{165}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{51 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{684}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{7}{139}-\frac{686 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{2035 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}-\frac{998}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{108}{139}-\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\frac{109 \left(\mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-2\right) \left(\mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{364}{109}\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)}{139}-\frac{1408}{139}-\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\frac{109 \left(\mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-2\right) \left(\mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{364}{109}\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}}{139}+\left(\frac{4078 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{582 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{5972}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{10715}{139}+\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{5972 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{1408}{139}-\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{10715}{139}+\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{5972 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{10715 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{1408 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{12726}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{388}{139}+\frac{179 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{388 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1232}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{81}{139}-\frac{258 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{81 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{1914}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{138}{139}+\frac{261 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{138 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1188}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{248 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{2200}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{16}{139}+\frac{2200 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{218}{139}-\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{165}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{218}{139}-\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1575}{139}+\frac{582 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{2422}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{165}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{728 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{2422}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{5864}{139}+\frac{1408 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{388}{139}+\frac{179 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{388 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1232}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{457}{139}+\frac{38 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{457 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{286}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{677 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1738}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{3836}{139}+\frac{1738 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{218}{139}+\frac{109 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{847}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{108 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{165}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{1014}{139}-\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{387}{139}+\frac{420 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{387 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{2310}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\frac{387 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{2310}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{996}{139}+\frac{2310 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{538}{139}+\frac{296 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{538 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1628}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\frac{538 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1628}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{2824}{139}+\frac{1628 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{170}{139}-\frac{70 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{170 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{385}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{170 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{385}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1010}{139}-\frac{385 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{20 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{1012}{139}+\frac{110 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{847}{139}+\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{1014}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{110}{139}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{847}{139}-\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}-\frac{1014}{139}\right) \left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-2\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{847}{139}+\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{2708}{139}-\frac{271 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2787 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{5298}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{13902}{139}+\frac{6027 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{1014}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{330 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}-\frac{2028}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{13902}{139}+\frac{6027 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{12369 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{139}+\frac{39513}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{165}{139}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}-\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2422}{139}+\frac{1575 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{5864}{139}-\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{2422 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{2422}{139}+\frac{1575 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{218 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{10781 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{1575 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{16735}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{24097}{139}-\frac{2422 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{16735 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{5864}{139}-\frac{165 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}-\frac{2422 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{24097}{139}-\frac{2422 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{16735 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{24097 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}+\frac{5864 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{32634}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1012}{139}-\frac{110 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{220}{139}-2 \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{220 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{2024}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{110}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{1012}{139}+\frac{110 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{6479}{139}-\frac{2239 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{6479 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{12067}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{6192 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{12888}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{37896}{139}-\frac{12888 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{6589}{139}-\frac{2378 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{6589 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{13079}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{6589 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}-\frac{13079}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{40822}{139}-\frac{13079 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}+\frac{6963 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{139}\right)}{489731}\)

This specification was found using the strategy pack "Point Placements" and has 97 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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\(\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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \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_{15}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= x^{2}\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \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_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{48}\! \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_{41}\! \left(x \right)\\ F_{61}\! \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_{64}\! \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_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \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_{76}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{71}\! \left(x \right)+F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{66}\! \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_{93}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{66}\! \left(x \right)\\ \end{align*}\)