Av(1234, 1324, 1342, 2413, 3214)
Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{3}}{x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 146, 394, 1075, 2949, 8089, 22159, 60668, 166107, 454864, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-x^{4}-4 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-5 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-5 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{1445304521 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{1586}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1586 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}+\frac{2894}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{1586 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}+\frac{2894}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{2894 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}-\frac{6595}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{1586 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}+\frac{2894}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{2894 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}-\frac{6595}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{2894 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}-\frac{6595}{5911}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6595 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{5911}-\frac{5171}{5911}\right) \left(\left(\left(\left(-1+\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{28223}{12869}-\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{1272}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1303217}{244511}-\frac{579259 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{101668}{12869}+\frac{784412 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{37936 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{136304}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{28223}{12869}-\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{101668}{12869}+\frac{784412 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{319962 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{1086558}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{23345 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{14383}+\frac{814667}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{1272}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{37936 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{136304}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{23345 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{14383}+\frac{814667}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{926803 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{2132121}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{13768}{244511}-\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{21691}{244511}+\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{511854 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{17671}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{21691}{244511}+\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{511854 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1628252 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{511854 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{495350}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{495350 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{981007}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{17671}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{495350 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{981007}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{17671 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{981007 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{2203996}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{766980}{244511}+\frac{344852 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{766980 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{1869588}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{455575}{244511}+\frac{156853 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{455575 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{742757}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1723915}{244511}-\frac{40201 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{14383}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1723915 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{4558345}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{365961 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{830131}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{830131 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{1683577}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{28223}{12869}-\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{1272}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{28223}{12869}-\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1392978}{244511}+\frac{511854 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{94465}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{1272}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{21691 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{94465}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{17671 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{71875}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{766980}{244511}+\frac{344852 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{766980 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{1869588}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{538714}{244511}-\frac{272558 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{538714 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{1388958}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{855 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{41839}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{41839 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{2119}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{28223}{12869}+\frac{234407 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}+\frac{72173}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{13768 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{1272}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{112136}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{1588516}{244511}+\frac{796332 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1588516 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{3442476}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{1588516 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{3442476}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{3442476 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{8419172}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{52331}{12869}-\frac{429411 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{52331 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{125395}{14383}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{52331 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{125395}{14383}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{125395 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{14383}+\frac{285723}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{230743}{244511}-\frac{110445 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{230743 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{498301}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{230743 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{498301}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{498301 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{1234983}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{100341 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{522469 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{70901}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{72173}{12869}+\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}+\frac{112136}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{522469}{244511}+\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{72173}{12869}-\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}-\frac{112136}{244511}\right) \left(\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{72173}{12869}+\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1483423}{244511}-\frac{1735 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{757}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1629736 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{4472168}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{930470 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{2270018}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}+\frac{112136}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{12869}-\frac{112136}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{930470 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}-\frac{2270018}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1877138 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)}{244511}+\frac{4974038}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{1272}{12869}+\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}-\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{94465}{244511}-\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{12869}+\frac{1392978 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{12869}-\frac{94465 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{71875}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{94465}{244511}-\frac{28223 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{12869}+\frac{1392978 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{3211644 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{1392978 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}+\frac{1096810}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1096810 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{94465 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{1949013}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{1272 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{12869}-\frac{94465 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{71875}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1096810 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{94465 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{1949013}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{71875 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{244511}-\frac{1949013 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{268820}{14383}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{522469 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{70901}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{522469}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{522469 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{70901}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{2089876}{244511}-4 \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{2089876 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{283604}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{900844}{244511}+\frac{491044 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{900844 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{3068756}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{833798 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}-\frac{1821474}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1821474 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{264486}{14383}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{90079}{12869}-\frac{731511 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{90079 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{3666839}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{90079 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{12869}-\frac{3666839}{244511}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{3666839 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =3\right)}{244511}+\frac{492751}{12869}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}+\frac{757976 \mathit{RootOf}\left(Z^{5}-Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{244511}\right)}{71815952072}\)
This specification was found using the strategy pack "Point Placements" and has 66 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 66 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{14}\! \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_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
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_{23}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \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_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{31}\! \left(x \right)\\
\end{align*}\)