Av(1234, 1342, 1423, 2143, 2413)
Generating Function
\(\displaystyle -\frac{x^{3}-2 x^{2}+3 x -1}{x^{5}-3 x^{3}+4 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 169, 503, 1501, 4480, 13368, 39886, 119009, 355095, 1059522, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+x^{3}-2 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{103916 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{917}{1252}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{917 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1252}+\frac{189}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{917 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1252}+\frac{189}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{189 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{626}+\frac{323}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{917 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1252}+\frac{189}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{189 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{626}+\frac{323}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{189 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{626}+\frac{323}{626}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{323 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{626}-\frac{10289}{1252}\right) \left(\left(\left(\left(-1-\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8906}{1411}+\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{6097}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1-\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{18834}{1411}+\frac{573 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{101315}{1411}+\frac{7884 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8906}{1411}+\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{101315}{1411}+\frac{7884 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{12230}{1411}-\frac{83887 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{44386}{1411}+\frac{69439 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{6097}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{44386}{1411}+\frac{69439 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{107647}{1411}-\frac{44386 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{1022}{1411}+\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{83}-\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{63}{1411}-\frac{70 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}+\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{694}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{63}{1411}-\frac{70 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{152496}{1411}-\frac{191110 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{70 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{74042}{1411}+\frac{152496 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}+\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{694}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{74042}{1411}+\frac{152496 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{107647}{1411}-\frac{74042 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{694 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6677}{83}-\frac{584 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{252 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}-3\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1752}{83}-3 \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{71838 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{96157}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{277850}{1411}+\frac{96157 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{62677}{1411}+\frac{66373 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{352654}{1411}-\frac{62677 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1-\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8906}{1411}+\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{6097}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8906}{1411}+\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{70 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}+\frac{107349}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{82363}{1411}-\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{6097}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{82363}{1411}-\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{29656}{1411}-\frac{694 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6677}{83}-\frac{584 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{7814 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{6034}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{694}{1411}+\frac{6034 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{694}{1411}+\frac{6034 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{29656}{1411}-\frac{694 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1984 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{8906}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{107412}{1411}-\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{6097}{1411}+\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{81669}{1411}+\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{73317 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{60729}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{22965}{1411}+\frac{60729 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{22965}{1411}+\frac{60729 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{101382}{1411}-\frac{22965 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{3530 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{10267}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{30478}{1411}+\frac{10267 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{30478}{1411}+\frac{10267 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{310871}{1411}-\frac{30478 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{3395 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{1022}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{6097}{1411}+\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{6097}{1411}+\frac{1022 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{81669}{1411}+\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-\frac{84 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{584}{83}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{584}{83}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{6677}{83}+\frac{584 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{6097}{1411}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}-\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{82363}{1411}-\frac{107349 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}+\frac{82363 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{29656}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{82363}{1411}-\frac{107349 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{60725}{1411}+\frac{222977 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{107349 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{55603}{1411}-\frac{60725 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{82363 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}+\frac{82363 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{29656}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{55603}{1411}-\frac{60725 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}+\frac{82363 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{902249}{1411}-\frac{55603 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{29656 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{584}{83}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{107412}{1411}+\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{584}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{107412}{1411}-\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}+\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{107412}{1411}+\frac{8906 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{404236}{1411}+\frac{58251 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{214485}{1411}-\frac{31069 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{6097 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}-\frac{81669}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{214485}{1411}-\frac{31069 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1004763}{1411}-\frac{55603 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{584}{83}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6677}{83}-\frac{584 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-3 \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{1752}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{20031}{83}+\frac{1752 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{83}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{84969 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{82000}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{522859}{1411}-\frac{82000 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{30522}{1411}-\frac{49360 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{652677}{1411}-\frac{30522 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{79325 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}-\frac{42288}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{68823}{1411}-\frac{42288 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{68823}{1411}-\frac{42288 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1116641}{1411}-\frac{68823 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1411}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}-\frac{52519 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{83}\right)}{2758245361}\)
This specification was found using the strategy pack "Point Placements" and has 93 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_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \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_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{47}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{47}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \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_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{63}\! \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_{74}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \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_{2}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{66}\! \left(x \right)+F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{58}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{64}\! \left(x \right)\\
\end{align*}\)