Av(1234, 1342, 1423, 2413, 3142)
View Raw Data
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
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{29241 \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) \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{56}{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)^{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{1454}{171}-\frac{310 \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)^{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{232 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{6491}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{196}{9}-\frac{91 \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} =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} =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{1454}{171}-\frac{310 \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{196}{9}-\frac{91 \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{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 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{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}{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{35 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{57}+\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{122}{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}{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}-\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} =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)^{2}}{57}+\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{122}{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{116 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{9}+\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{1061}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{910}{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{1061 \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{53}{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}-\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} =2\right)^{2}+\left(\frac{910}{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{1061 \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{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(1-\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)+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{293}{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(-\frac{220}{171}+\frac{179 \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} =4\right)-\frac{220 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{2467}{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{106}{57}-\frac{1367 \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{106 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+\frac{6815}{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{364}{57}+\frac{764 \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{364 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}-\frac{11642}{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{959 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{526}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{3341}{57}-\frac{526 \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)^{-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{56}{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)^{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{1766}{171}-\frac{142 \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{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{171}-\frac{1127}{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}{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)^{2}+\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{1127}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{63}{19}+\frac{53 \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} =3\right)^{-n +2}+\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} =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)+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{293}{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{220}{171}+\frac{179 \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} =4\right)-\frac{220 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{2467}{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{149}{57}-\frac{136 \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} =4\right)-\frac{149 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{57}+\frac{1697}{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{48 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}+\frac{109}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1475}{57}+\frac{109 \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} =2\right)^{-n +2}+\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} =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)+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{293}{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} =3\right)}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{548}{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{122 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\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} =4\right)+\frac{1286}{171}-\frac{56 \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)^{-n +3}+\left(\left(\left(\frac{409}{171}-\frac{1409 \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{409 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{202}{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{409 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{202}{19}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{2681}{171}+\frac{202 \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} =1\right)^{-n +1}+\left(\left(\left(-\frac{85}{19}+\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{85 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}-\frac{1724}{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{85 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}-\frac{1724}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{6691}{171}-\frac{1724 \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)^{-n +2}+\left(\left(\left(\frac{391}{171}-\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{391 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{823}{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{391 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{823}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{3478}{171}+\frac{823 \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)^{-n +3}+\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} =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)-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{293}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{293}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{1588}{171}-\frac{293 \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)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{293}{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{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{1286}{171}+\frac{56 \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)^{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{1286}{171}-\frac{56 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{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{854 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{57}+\frac{13576}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{6983}{171}+\frac{1679 \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{1286}{171}+\frac{56 \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{1286}{57}-\frac{56 \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{6983}{171}+\frac{1679 \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{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(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\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(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}-\frac{1766 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1127}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{63}{19}-\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{1127 \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(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}-\frac{1766 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{1127}{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{4990 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{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{1135}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{806}{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{1135 \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{63}{19}-\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{1127 \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)^{2}+\left(-\frac{806}{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{1135 \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{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(\mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{293}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1588}{171}+\frac{293 \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)^{4}+\left(\left(-\frac{293}{57}-3 \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)}{57}+\frac{1588}{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(\frac{2051}{171}+7 \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{2051 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}-\frac{11116}{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{1790}{171}-7 \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{1790 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{13087}{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{143 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{19}+\frac{673}{57}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{13358}{171}+\frac{673 \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} =2\right)^{-n}+\left(\left(\left(\frac{554}{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{554 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{383}{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{554 \mathit{RootOf}\left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{171}+\frac{383}{171}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}+7 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{5711}{171}+\frac{383 \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)^{-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)}{6848689}\)

This specification was found using the strategy pack "Point Placements" and has 59 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_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \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_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \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_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\ 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_{51}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{52}\! \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_{58}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ \end{align*}\)