Av(1243, 1342, 2134, 2143, 2413)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{x^{5}-3 x^{4}+7 x^{3}-8 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 180, 545, 1647, 4977, 15043, 45473, 137464, 415553, 1256212, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{4}+7 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{29241 \left(\left(\left(\left(-1+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-1-\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{56}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{49}{171}-\frac{214 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2213}{171}+\frac{14 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{9}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{310 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{1454}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-1-\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{2213}{171}+\frac{14 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{9}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{6491}{171}+\frac{232 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{91 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}+\frac{196}{9}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{56}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{310 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{1454}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{91 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}+\frac{196}{9}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1021}{171}-\frac{134 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{122}{171}+\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{122}{171}+\frac{142 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}-\frac{53}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{122}{171}+\frac{142 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1061}{171}+\frac{142 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}-\frac{116 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{9}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1061 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}+\frac{910}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}-\frac{53}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1061 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}+\frac{910}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{680}{171}-\frac{53 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{57}+\frac{910 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{293}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\frac{179 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{220}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{2467}{171}-\frac{220 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{1367 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{106}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{6815}{171}+\frac{106 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{364}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{11642}{171}+\frac{364 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{526}{171}-\frac{959 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{526 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{3341}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-1-\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{56}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-1-\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{142 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{1766}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1127}{171}+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{56}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1127}{171}+\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{53 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}-\frac{63}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{293}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{179 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{220}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{2467}{171}-\frac{220 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{136 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{149}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{1697}{57}-\frac{149 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{109}{57}+\frac{48 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{109 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{1475}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{293}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{548}{57}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{56}{171}-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1286}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{1409 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{409}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{202}{19}+\frac{409 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{202}{19}+\frac{409 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{202 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}-\frac{2681}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{959 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{85}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1724}{171}-\frac{85 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{1724}{171}-\frac{85 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1724 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{6691}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{48 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}+\frac{391}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{823}{171}+\frac{391 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{823}{171}+\frac{391 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{823 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{3478}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{293}{171}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{293}{171}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{293 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1588}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{293}{171}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{548}{57}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{1286}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}-\left(\left(\frac{293}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{548}{57}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}+\frac{1286}{171}\right) \left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-3\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{548}{57}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{6218}{171}+\frac{569 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{13576}{171}-\frac{854 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1679 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{6983}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{1286}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}+\frac{1286}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1679 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{6983}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{3472}{171}+\frac{895 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{56}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1127}{171}-\frac{1766 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1127 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}+\frac{63}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1127}{171}-\frac{1766 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1135}{171}-\frac{1766 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}+\frac{4990 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1135 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1127 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}-\frac{806}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{1127 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}+\frac{63}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1135 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1127 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{171}-\frac{806}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1012}{171}+\frac{63 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{19}-\frac{806 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{293}{171}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{293 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{1588}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(-3 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{293}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{1588}{57}-\frac{293 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(7 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{2051}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{11116}{171}+\frac{2051 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-7 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{1790}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{13087}{171}-\frac{1790 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{673}{57}+\frac{143 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{673 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{13358}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{554}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{383}{171}+\frac{554 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{383}{171}+\frac{554 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{383 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{5711}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}-\frac{2617 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{171}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{122}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{22}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{22}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{22 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+\frac{31}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{22}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{22 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+\frac{31}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{22 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+\frac{31}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{31 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{250}{171}\right)}{6848689}\)
This specification was found using the strategy pack "Point Placements" and has 69 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 69 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{21}\! \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_{14}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
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_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \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_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{40}\! \left(x \right)\\
\end{align*}\)