Av(1342, 1432, 2143, 3142, 3214)
Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{4}}{x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 163, 457, 1280, 3602, 10171, 28749, 81241, 229477, 648046, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+9 a \! \left(n +2\right)-9 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+9 a \! \left(n +2\right)-9 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{2279169 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{38}{471}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{38 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{471}-\frac{28}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{38 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{471}-\frac{28}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{28 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{157}+\frac{73}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{38 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{471}-\frac{28}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{28 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{157}+\frac{73}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{28 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{157}+\frac{73}{157}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{73 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{157}+\frac{619}{471}\right) \left(\left(\left(\left(-1+\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2921}{1613}-\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{1570}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{30869}{4839}-\frac{7709 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{75412}{4839}+\frac{22204 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{5092 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{3300}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2921}{1613}-\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{75412}{4839}+\frac{22204 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{54560}{1613}-\frac{4832 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{5277 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}-\frac{29045}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{1570}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{5092 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{3300}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{5277 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}-\frac{29045}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{18425}{4839}-\frac{13475 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}+\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{1174}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{6334 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}+\frac{2617}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{327}{1613}+\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{6334 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}+\frac{2617}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{6334 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}+\frac{276}{1613}+\frac{19970 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{276 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{15125}{4839}+\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{327}{1613}+\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{276 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{15125}{4839}+\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{32204}{4839}-\frac{15125 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{327 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{7589}{4839}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\frac{6346 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{22106}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{68750}{4839}-\frac{22106 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{19038 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{22106}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{68750}{1613}+\frac{22106 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{53686 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-14\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{414502}{4839}-14 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{655}{4839}+\frac{185 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{655 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{28285}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2921}{1613}-\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{1570}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2921}{1613}-\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{6334 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{50170}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{2057}{1613}-\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{1570}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2057}{1613}-\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{327 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}+\frac{1531}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{7589}{4839}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{6346 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{22106}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{68750}{4839}-\frac{22106 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{5290 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{8414}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{47226}{1613}+\frac{8414 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{1243}{1613}+\frac{825 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1243 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{6721}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{7589}{4839}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1363 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{2921}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{15851}{1613}-\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{1570}{4839}-\frac{1174 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{1730}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{32462 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{14702}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{67646}{4839}+\frac{14702 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{67646}{4839}+\frac{14702 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{67646 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{385562}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{13748 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{13692}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{21524}{1613}-\frac{13692 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{21524}{1613}-\frac{13692 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{21524 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{131228}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{4983 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{13343}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{21197}{4839}+\frac{13343 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{21197}{4839}+\frac{13343 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{21197 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{44253}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{4733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{7589}{4839}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{7589}{4839}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{7589 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{45983}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{7589}{4839}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{15851}{1613}+\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{1730}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{7589}{4839}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{15851}{1613}-\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}-\frac{1730}{1613}\right) \left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-3\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{15851}{1613}+\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{49283}{1613}-\frac{27859 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{398870}{4839}+\frac{22982 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{18292 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{51484}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{1730}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{1613}-\frac{5190}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{18292 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}+\frac{51484}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{20156}{1613}+\frac{12284 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}-\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{1570}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{50170 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}-\frac{2057}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1531}{1613}-\frac{2057 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{50170 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{2921 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}-\frac{2057}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{50170 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}-\frac{4372}{4839}-\frac{31302 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{4372 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{26063}{4839}-\frac{2057 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{1531}{1613}-\frac{2057 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{1570 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4372 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{26063}{4839}-\frac{2057 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{33932}{1613}+\frac{26063 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{1531 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{7589}{4839}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{7589 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}+\frac{45983}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(-3 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{7589}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{45983}{1613}+\frac{7589 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{22767}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{137949}{1613}-\frac{22767 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4877 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}+\frac{44677}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{375967}{4839}+\frac{44677 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{4774}{4839}-\frac{4162 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{4774 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{20434}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{9640 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1613}-\frac{23624}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{37880}{4839}-\frac{23624 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{37880}{4839}-\frac{23624 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{37880 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)}{4839}-\frac{80920}{1613}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}-\frac{82016 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{4839}\right)}{420414016}\)
This specification was found using the strategy pack "Point Placements" and has 57 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 57 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_{18}\! \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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
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_{27}\! \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_{23}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \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_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \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_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{54}\! \left(x \right)\\
\end{align*}\)