Av(1243, 1324, 1342, 3124, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{5}-x^{4}+3 x^{3}-4 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 173, 513, 1521, 4512, 13388, 39727, 117884, 349801, 1037973, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-x^{4}+3 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{207507 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{555}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{41}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{41}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{41 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{469}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{41}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{41 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{469}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{41 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{469}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{469 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{779}{263}\right) \left(\left(\left(\left(\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{4}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{781}{263}+\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{2770 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{2980}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{5885}{789}+\frac{1484 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{7174}{789}+\frac{226 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{4}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{5885}{789}+\frac{1484 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{8134}{789}+\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{271 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}+\frac{7801}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{781}{263}+\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{7174}{789}+\frac{226 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{271 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}+\frac{7801}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{3350}{789}+\frac{990 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}+\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{555}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{740}{263}+\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{789}+\frac{325 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1243}{789}-\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{740}{263}+\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{789}+\frac{325 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3752}{789}+\frac{325 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{789}-\frac{8365 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{3752 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{656}{263}-\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{1243}{789}-\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3752 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{656}{263}-\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1105}{789}-\frac{1243 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{789}+\frac{656 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2717}{789}+\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\frac{633 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{1928}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2864}{789}+\frac{1928 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{1373 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{350}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2766}{263}-\frac{350 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1993 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{1562}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{1500}{263}+\frac{1562 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{1353}{263}-\frac{1155 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{1353 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{11143}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{4}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{781}{263}+\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{4}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{325 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}+\frac{1570}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1196}{263}+\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{781}{263}+\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1196}{263}+\frac{740 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{1243 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}+\frac{2245}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2717}{789}+\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{633 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{1928}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2864}{789}+\frac{1928 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{603 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{1175}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{3733}{789}-\frac{1175 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{3586}{789}+\frac{514 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{3586 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{862}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2717}{789}+\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{871 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{4}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{830}{263}+\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{3}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{781}{263}-\frac{555 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}+\frac{4831}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{4814 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{103}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{6809}{789}-\frac{103 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{6809}{789}-\frac{103 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{6809 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{10555}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{770 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{275}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{4565}{789}-\frac{275 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{4565}{789}-\frac{275 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{4565 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{715}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{1028 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{292}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{374}{789}-\frac{292 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{374}{789}-\frac{292 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{374 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{4684}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{370 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2717}{789}-\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{2717}{789}-\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2717 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{49}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{2717}{789}-\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{3}+\frac{830}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{4831}{789}+\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}-\left(\left(\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2717}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{3}-\frac{830}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{4831}{789}-\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \left(\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-1\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{3}+\frac{830}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{3395 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}-\frac{7321}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{5498}{789}-\frac{8129 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{4380 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}+\frac{1795}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{4831}{789}+\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{4831}{789}-\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{4380 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{263}+\frac{1795}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{2831}{789}+\frac{338 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}+\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{3}-\frac{781}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1196}{263}+\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{3}-\frac{1570 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{2245}{789}+\frac{1196 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1196}{263}+\frac{4 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{3}-\frac{1570 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{7307}{789}-\frac{1570 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}+\frac{3757 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{7307 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{861}{263}+\frac{1196 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{2245}{789}+\frac{1196 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}-\frac{781 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{7307 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{861}{263}+\frac{1196 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{13108}{789}-\frac{2245 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{789}+\frac{861 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{2717}{789}+\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2717 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{49}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{2717}{789}-\mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2717 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}+\frac{49}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(3 \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{2717}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{147}{263}+\frac{2717 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{10 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{3}-\frac{6803}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{5219}{789}-\frac{6803 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{16288}{789}+\frac{2037 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{16288 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{3706}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{2 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{3}+\frac{1355}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{4631}{789}+\frac{1355 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{4631}{789}+\frac{1355 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{263}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{4631 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{789}-\frac{13247}{789}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}-\frac{5038 \mathit{RootOf}\left(Z^{5}-Z^{4}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{789}\right)}{25381444}\)
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This specification was found using the strategy pack "Point Placements" and has 51 rules.

Found on January 18, 2022.

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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_{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_{37}\! \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_{24}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \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_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{49}\! \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_{46}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{15}\! \left(x \right)\\ \end{align*}\)