Av(1234, 1342, 1432, 2314)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{5}-6 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 190, 564, 1672, 4968, 14785, 44023, 131079, 390249, 1161783, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = 6 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = 6 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{289728 \left(\left(\left(\left(\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}-1\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{3519}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{19442}{1509}-\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}-1\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1061 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{587}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{4985 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}+\frac{32569}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{28841}{1006}+\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{3519}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{4985 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}+\frac{32569}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{34696 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{26568}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{24295 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{71171}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{19442}{1509}-\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{28841}{1006}+\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{24295 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{71171}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{109049}{1006}+\frac{71171 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{503}+\frac{6451}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}+\frac{10931 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{7341}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}-\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{7935}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}+\frac{10931 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{7341}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{44500 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}+\frac{10931 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}+\frac{10208}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{503}+\frac{10208 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{61753}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}-\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{7935}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{503}+\frac{10208 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{61753}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{109049}{1006}+\frac{7935 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}+\frac{61753 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1466 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{2105}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-6+\frac{1416 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-6 \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)+\frac{8796}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{45409}{1006}+\frac{30674 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{45409 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{149933}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{63001 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{187823}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{187823 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{282095}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}-1\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{3519}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{19442}{1509}-\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}-\frac{3519}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{35150}{1509}-\frac{10931 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}-\frac{18388}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{19442}{1509}-\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{7341 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}-\frac{18388}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{7935 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{9418}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1466 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{2105}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2581}{1509}-\frac{961 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2581 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{7935}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{2581 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{7935}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{7935 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{9418}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{3519}{1006}-\frac{2570 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{13127}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{19442}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}+\frac{28841}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{71049}{1006}+\frac{45392 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{71049 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{223007}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{71049 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{223007}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{223007 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{315775}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{11635}{1509}-\frac{3793 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{11635 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{25527}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{11635 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{25527}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{25527 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{47308}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{6451}{1006}-\frac{4079 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{19442}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{6451 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{19442}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}+\frac{28841}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{236 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)+\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)+\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1466 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}-\frac{2105}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-\frac{19442}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}-\frac{35150 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}+\frac{18388}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}+\frac{18388 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}-\frac{9418}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1006}-\frac{35150 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}+\frac{18388}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{70201 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}-\frac{35150 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}-\frac{13730}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{18388 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{503}-\frac{13730 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{26615}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}+\frac{18388 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}-\frac{9418}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{18388 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{503}-\frac{13730 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{26615}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{228638}{1509}-\frac{9418 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1509}-\frac{26615 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{13127}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{28841}{1006}+\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{1466}{503}+\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}-\frac{13127}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{28841}{1006}-\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{3519 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{13127}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{28841}{1006}+\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{3351 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1006}+\frac{5231}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{4312 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}+\frac{45661}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{28841}{1006}+\frac{19442 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{4312 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1509}+\frac{45661}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{170591}{1006}-\frac{26615 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{1466}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1466 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}+\frac{2105}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-6 \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)+\frac{8796}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{8796 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{503}-\frac{12630}{503}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{83993}{1509}-\frac{24465 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{83993 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{107359}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{112340 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{127385}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{127385 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{514445}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{119177}{1509}-\frac{32513 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{119177 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{141039}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{119177 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1509}-\frac{141039}{1006}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{141039 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1006}+\frac{541622}{1509}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}+\frac{231917 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{3018}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{95}{16}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{95 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{16}-\frac{757}{96}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{95 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{16}-\frac{757}{96}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{757 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{96}+\frac{101}{4}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{95 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{16}-\frac{757}{96}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{757 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{96}+\frac{101}{4}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{757 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{96}+\frac{101}{4}\right) \mathit{RootOf} \left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{101 \mathit{RootOf}\left(Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4}+\frac{2173}{96}\right)}{7683642127}\)
This specification was found using the strategy pack "Point Placements" and has 158 rules.
Found on January 18, 2022.Finding the specification took 5 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_{11}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \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_{43}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \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_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{84}\! \left(x \right)+F_{94}\! \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_{56}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)+F_{54}\! \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_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{53}\! \left(x \right) &= 0\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{57}\! \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_{70}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 0\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{78}\! \left(x \right)+F_{82}\! \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_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{87}\! \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_{2}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{91}\! \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_{87}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{22}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{112}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= x^{2}\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{145}\! \left(x \right)+F_{150}\! \left(x \right)+F_{22}\! \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_{127}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{124}\! \left(x \right)+F_{125}\! \left(x \right)+F_{22}\! \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_{123}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 0\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{133}\! \left(x \right)+F_{142}\! \left(x \right)+F_{143}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \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_{136}\! \left(x \right)\\
F_{142}\! \left(x \right) &= 0\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{77}\! \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_{136}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{97}\! \left(x \right)\\
\end{align*}\)