Av(1234, 1324, 1342, 1432, 2413)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(2 x -1\right)^{2}}{2 x^{5}-3 x^{3}+7 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 168, 494, 1453, 4273, 12562, 36922, 108507, 318861, 936976, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-3 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-\left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n \right) = \frac{3 a \! \left(n +2\right)}{2}-\frac{7 a \! \left(n +3\right)}{2}+\frac{5 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{18622387 \left(\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{2266}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{2266 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}+\frac{1369}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{2266 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}+\frac{1369}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1369 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}-\frac{324}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{2266 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}+\frac{1369}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1369 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}-\frac{324}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{1369 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}-\frac{324}{3703}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{324 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3703}-\frac{67}{161}\right) \left(\left(\left(\left(-1+\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{337}{5029}-\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{1874}{5029}-\frac{13266 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{32131}{10058}+\frac{2885 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{337}{5029}-\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{32131}{10058}+\frac{2885 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{63398}{5029}-\frac{88199 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{51738 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{29761}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{51738 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{29761}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{371}{107}-\frac{29761 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{2548}{5029}-\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}+\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{15206}{5029}+\frac{21337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{9551}{5029}-\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{15206}{5029}+\frac{21337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{59180}{5029}-\frac{232197 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}+\frac{21337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{59180 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}-\frac{56501}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{9551}{5029}-\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{59180 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}-\frac{56501}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{9551 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}+\frac{371}{107}-\frac{56501 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{38743}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{3}{2}+\frac{10113 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{3 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2}-\frac{6633}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{104487}{10058}-\frac{204415 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{104487 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{225355}{20116}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{55560 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{54563}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{54563 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}+\frac{34927}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{337}{5029}-\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{337}{5029}-\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{5025}{10058}-\frac{21337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{2109}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{15206 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{2109}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{9551 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{3021}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{38743}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{13553}{5029}-\frac{24222 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{13553 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{9551}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{13553 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{9551}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{9551 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{3021}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{337}{5029}+\frac{8237 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{35437}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{75917}{10058}-\frac{71067 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{35020}{5029}+\frac{75917 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{35020}{5029}+\frac{75917 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{35020 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{131493}{20116}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{42193}{10058}-\frac{34335 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{12469}{10058}+\frac{42193 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{12469}{10058}+\frac{42193 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{12469 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}+\frac{110187}{20116}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{2548}{5029}+\frac{3208 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{2548 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{1653}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{6742 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{2211}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{2211}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{2211 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{38743}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{1653}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2109}{5029}+\frac{5025 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{3021}{10058}-\frac{2109 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{2109}{5029}+\frac{5025 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{76565}{10058}+\frac{64230 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{5025 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{76565 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{2109 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}+\frac{25165}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{3021}{10058}-\frac{2109 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}-\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{76565 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{2109 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{5029}+\frac{25165}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{3021 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{10058}-\frac{8511}{10058}+\frac{25165 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{2211}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{35437}{10058}+\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{35437}{10058}-\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{35437}{10058}+\frac{337 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{24817}{20116}+\frac{68448 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{39793 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{5753}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{1653 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}+\frac{11660}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{39793 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}-\frac{5753}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{38669}{20116}+\frac{25165 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{2211}{5029}+\mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2211 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}+\frac{38743}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{3 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2}-\frac{6633}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{116229}{20116}-\frac{6633 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{40747}{10058}+\frac{137907 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{361823}{20116}-\frac{40747 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{29227}{10058}-\frac{74627 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{29227 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}-\frac{146197}{20116}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{28112}{5029}+\frac{51352 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{45311}{10058}-\frac{28112 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{45311}{10058}-\frac{28112 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{45311 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{10058}+\frac{12985}{5029}\right) \mathit{RootOf} \left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}+\frac{7261 \mathit{RootOf}\left(2 Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{5029}\right)}{52722121}\)

This specification was found using the strategy pack "Point Placements" and has 99 rules.

Found on January 18, 2022.

Finding the specification took 4 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_{16}\! \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_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{29}\! \left(x \right)\\ 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_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{65}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)+F_{63}\! \left(x \right)+F_{64}\! \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_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{40}\! \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_{47}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{59}\! \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_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 0\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 0\\ F_{64}\! \left(x \right) &= 0\\ 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_{47}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{69}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{77}\! \left(x \right)+F_{88}\! \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_{81}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{63}\! \left(x \right)+F_{83}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{87}\! \left(x \right) &= 0\\ 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_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{68}\! \left(x \right)\\ \end{align*}\)