Av(1234, 1324, 1342, 2143, 3124)
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Generating Function
\(\displaystyle -\frac{\left(x^{2}-x +1\right) \left(x^{2}+2 x -1\right)}{x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 175, 529, 1603, 4860, 14732, 44651, 135330, 410169, 1243181, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x^{2}-x +1\right) \left(x^{2}+2 x -1\right) = 0\)
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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{776251 \left(\left(\left(\left(-\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-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{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{5916}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{2561}{1211}\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{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-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{3504 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{12184}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{447 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{173}+\frac{507}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1702 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{14852}{1211}\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{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{5916}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{447 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{173}+\frac{507}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{21593 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{4643}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{25304}{1211}-\frac{838 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{2561}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1702 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{14852}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{25304}{1211}-\frac{838 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{7891 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-\frac{60455}{1211}\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{859}{1211}+\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{695}{1211}-\frac{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}+\frac{4539 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{6708}{1211}\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{695}{1211}-\frac{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}+\frac{4539 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{35121 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{4539 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{9168}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}+\frac{27043}{1211}-\frac{9168 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{6708}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}+\frac{27043}{1211}-\frac{9168 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{27043 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{102097}{1211}+\frac{6708 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}\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(1+\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{5057}{1211}\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{6268}{1211}+\frac{142 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6268 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{102}{173}\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{1424}{1211}-\frac{4418 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1424 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{19514}{1211}\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{14443}{1211}-\frac{6441 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{14443 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{12821}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{104 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}-\frac{208}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{8216}{1211}-\frac{208 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-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{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{5916}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{2561}{1211}\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{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{5916}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{7599}{1211}-\frac{4539 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{10705}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{2561}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{695 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{10705}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{41642}{1211}-\frac{6708 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}\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(1+\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{5057}{1211}\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{6268}{1211}+\frac{142 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{6268 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{102}{173}\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{4050}{1211}-\frac{7668 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{4050 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{28}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{1007 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{4147}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{24229}{1211}-\frac{4147 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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(1+\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{5057}{1211}\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{5916}{1211}-\frac{3646 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{6904}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{859 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{2561}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{17413}{1211}-\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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{796}{173}-\frac{1012 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{796 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}+\frac{2860}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{796 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}+\frac{2860}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1308}{173}+\frac{2860 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}\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{2626}{1211}-\frac{3250 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{2626 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{19318}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{2626 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{19318}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{37258}{1211}-\frac{19318 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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{352}{1211}-\frac{3788 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{352 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{7618}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{352 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{7618}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{3108}{173}-\frac{7618 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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(-1-\frac{1069 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{4343}{1211}+\frac{5057 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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(-\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{6904}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{17413}{1211}\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{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-\frac{6904}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-\frac{17413}{1211}\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{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{6904}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{3355 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{10509}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{23116 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-\frac{53266}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{72677}{1211}-\frac{8509 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{17413}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}-\frac{17413}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{72677}{1211}-\frac{8509 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{1901 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{1211}+\frac{24233}{1211}\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(-\frac{2561}{1211}+\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}-\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{10705}{1211}-\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{7599 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{10705 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{41642}{1211}\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{10705}{1211}-\frac{5916 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{7599 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{36927 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{7599 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}+\frac{43838}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{10705 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{43543}{1211}+\frac{43838 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{2561 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{10705 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{41642}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{10705 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}-\frac{43543}{1211}+\frac{43838 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{43543 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{146303}{1211}+\frac{41642 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{1211}\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(\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)-\frac{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{4343}{1211}-\frac{5057 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{5057}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{4343}{1211}+\frac{5057 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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{15171}{1211}-3 \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{15171 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}+\frac{13029}{1211}\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{2163}{173}-\frac{401 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{2163 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}-\frac{56827}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{118 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}-\frac{20438}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{56522}{1211}-\frac{20438 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\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{5087}{1211}-\frac{1093 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{173}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{5087 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{39455}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{5087 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}-\frac{39455}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{78951}{1211}-\frac{39455 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{1211}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}-\frac{46826 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{1211}\right) \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{570}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{570 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}-\frac{1429}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{570 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}-\frac{1429}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1429 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}+\frac{3295}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\left(\frac{570 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}-\frac{1429}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1429 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}+\frac{3295}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{1429 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}+\frac{3295}{641}\right) \mathit{RootOf} \left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{3295 \mathit{RootOf}\left(Z^{5}-Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{641}-\frac{8466}{641}\right)}{1096337138}\)
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This specification was found using the strategy pack "Point Placements" and has 49 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{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_{43}\! \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_{26}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \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_{35}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right)\\ \end{align*}\)