Av(1243, 1342, 1423, 1432, 2314)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right)^{2}}{x^{5}-3 x^{3}+7 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 174, 519, 1545, 4595, 13659, 40591, 120608, 358335, 1064597, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-\left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right)-a \! \left(n +5\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{177555665 \left(\left(\left(\left(-1+\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{6324}{4655}-\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{187}{133}+\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{94784}{32585}-\frac{65522 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{159148}{32585}-\frac{7604 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6517}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{167354}{32585}+\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{6324}{4655}-\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{159148}{32585}-\frac{7604 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6517}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{414976}{32585}-\frac{49643 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{247622 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}-\frac{8011}{1715}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{187}{133}+\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{167354}{32585}+\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{247622 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}-\frac{8011}{1715}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{135974}{32585}-\frac{8011 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1715}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}+\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{6248}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{159146 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{118372}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{75094}{32585}-\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{159146 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{118372}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{159146 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}+\frac{230456}{32585}-\frac{497654 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{230456 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{91383}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{75094}{32585}-\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{230456 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{91383}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{135974}{32585}+\frac{75094 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{91383 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{250778}{32585}+\frac{50516 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{114474 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-3\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{151548}{32585}-3 \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{492127 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{114818}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{767098}{32585}+\frac{114818 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{14764}{32585}+\frac{266366 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{14764 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{91148}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{6324}{4655}-\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{187}{133}+\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{6324}{4655}-\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{159146 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}-\frac{86591}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{2636}{931}+\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{187}{133}+\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2636}{931}+\frac{118372 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{75094 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}-\frac{60826}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{250778}{32585}+\frac{50516 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{121126 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{72557}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{75094}{32585}+\frac{72557 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{75094}{32585}+\frac{72557 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{75094 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{60826}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{32937 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{6324}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{204963}{32585}+\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{187}{133}-\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{133}-\frac{167354}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{127266 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{38271}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{368376}{32585}+\frac{38271 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{368376}{32585}+\frac{38271 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{368376 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{31926}{931}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{2480 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{343}+\frac{170312}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{10922}{4655}+\frac{170312 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{10922}{4655}+\frac{170312 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{10922 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4655}+\frac{691508}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{352 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{6248}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{187}{133}-\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{187}{133}-\frac{6248 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{133}-\frac{167354}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{38158 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{50516}{32585}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{50516}{32585}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{50516 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{250778}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}+\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4655}-\frac{187}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{86591 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{4655}-\frac{2636}{931}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{60826}{32585}-\frac{2636 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{931}-\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{86591 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{4655}-\frac{2636}{931}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{86591 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}-\frac{295789}{32585}+\frac{355237 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{295789 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{2636 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{931}+\frac{274431}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{60826}{32585}-\frac{2636 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{931}-\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{295789 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{2636 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{931}+\frac{274431}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{79178}{32585}+\frac{60826 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{32585}+\frac{274431 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{50516}{32585}-\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{204963}{32585}-\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{167354}{32585}+\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{50516}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{204963}{32585}+\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{167354}{32585}-\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{204963}{32585}-\frac{6324 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4655}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{167354}{32585}+\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{874098}{32585}+\frac{539757 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{10189 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{931}-\frac{579224}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{167354}{32585}+\frac{187 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{133}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{10189 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{931}-\frac{579224}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{109131}{6517}+\frac{274431 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{50516}{32585}+\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{250778}{32585}+\frac{50516 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{151548}{32585}-3 \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{752334}{32585}-\frac{151548 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{406953 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{259209}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1963449}{32585}+\frac{259209 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{77162}{32585}-\frac{6262 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{931}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{77162 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{861146}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{178858 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}-\frac{94403}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{208003}{32585}-\frac{94403 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{208003}{32585}-\frac{94403 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{208003 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{32585}+\frac{88652}{6517}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}+\frac{143271 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{32585}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{3850}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{3850 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}+\frac{2431}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{3850 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}+\frac{2431}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2431 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}-\frac{216}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{3850 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}+\frac{2431}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2431 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}-\frac{216}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{2431 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}-\frac{216}{5449}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{216 \mathit{RootOf}\left(Z^{5}-3 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)}{5449}-\frac{4139}{5449}\right)}{2280731049}\)
This specification was found using the strategy pack "Point Placements" and has 69 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 69 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \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_{49}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= x^{2}\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
\end{align*}\)