Av(13425, 13452, 14325, 14352, 31425, 31452, 34125, 34152, 41325, 41352, 43125, 43152)
Generating Function
\(\displaystyle -\frac{\left(7 x^{2}-6 x +1\right) \left(x^{2}+2 x -1\right)}{6 x^{5}-6 x^{4}-21 x^{3}+25 x^{2}-9 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 504, 2364, 11052, 51456, 238824, 1106268, 5118516, 23667672, 109401984, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{5}-6 x^{4}-21 x^{3}+25 x^{2}-9 x +1\right) F \! \left(x \right)+\left(7 x^{2}-6 x +1\right) \left(x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)} + 21 a{\left(n + 2 \right)} - 25 a{\left(n + 3 \right)} + 9 a{\left(n + 4 \right)}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)} + 21 a{\left(n + 2 \right)} - 25 a{\left(n + 3 \right)} + 9 a{\left(n + 4 \right)}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{78231150 \left(\left(\left(\left(\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)-\frac{1889}{4164}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{1889 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{4164}+\frac{241}{1041}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(-\frac{1889 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{4164}+\frac{241}{1041}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{241 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1041}-\frac{481}{2776}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{1889 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{4164}+\frac{241}{1041}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{241 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1041}-\frac{481}{2776}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(\frac{241 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1041}-\frac{481}{2776}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{481 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{2776}+\frac{770}{3123}\right) \left(\left(\left(\left(-1+\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{656}{2505}-\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{847}{7515}+\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{6436}{7515}-\frac{95308 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{12525}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{58853}{37575}+\frac{51809 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{12525}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{7816}{12525}-\frac{4168 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{656}{2505}-\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{58853}{37575}+\frac{51809 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{12525}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{47746}{37575}+\frac{175834 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{112725}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{49249}{112725}-\frac{6637 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{847}{7515}+\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{7816}{12525}-\frac{4168 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{49249}{112725}-\frac{6637 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{64}{835}+\frac{1352 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{4509}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{37575}+\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-\frac{1003}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{261449 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}+\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{37575}+\frac{3437}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{159}{1670}-\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{1503}+\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{261449 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}+\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{37575}+\frac{3437}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{261449 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{75150}-\frac{187601}{75150}+\frac{5747 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{675}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{97244}{112725}-\frac{187601 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}+\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(-\frac{159}{1670}-\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{1503}+\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{97244}{112725}-\frac{187601 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}+\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{40901}{75150}-\frac{159 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{1670}+\frac{97244 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{112725}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}+1\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{3047}{7515}+\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{165412 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-\frac{4468}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{42248}{37575}-\frac{4468 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{819709 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}-\frac{58699}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{221611}{75150}-\frac{58699 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{997502 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{112725}+\frac{122}{835}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{525304}{112725}+\frac{122 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{835}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{31307}{45090}+\frac{4705 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{3006}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{188021}{75150}+\frac{31307 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{45090}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{656}{2505}-\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{847}{7515}+\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{656}{2505}-\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{37575}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{261449 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{75150}-\frac{158051}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{62521}{75150}-\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{847}{7515}+\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{62521}{75150}-\frac{3437 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{5010}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{35141}{75150}+\frac{159 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1670}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}+1\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{3047}{7515}+\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{165412 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-\frac{4468}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{42248}{37575}-\frac{4468 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{9881 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{15030}-\frac{8069}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{99341}{75150}-\frac{8069 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{625}{3006}-\frac{395 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{3006}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{809}{3006}+\frac{625 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{3006}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}+1\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{3047}{7515}+\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{120512 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-\frac{656}{2505}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{53248}{37575}-\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{2505}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(-\frac{847}{7515}-\frac{1003 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{27683}{37575}-\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{331666 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-\frac{3910}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{7478}{7515}-\frac{3910 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(-\frac{7478}{7515}-\frac{3910 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{145502}{37575}-\frac{7478 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{434557 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}+\frac{5063}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{12227}{7515}+\frac{5063 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(\frac{12227}{7515}+\frac{5063 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{211303}{37575}+\frac{12227 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{1796 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}+\frac{500}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{440}{1503}+\frac{500 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(\frac{440}{1503}+\frac{500 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{1192}{1503}+\frac{440 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{127837 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}-1\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{3047}{7515}-\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(-\frac{3047}{7515}-\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{6387}{4175}-\frac{3047 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{3047}{7515}-\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{53248}{37575}+\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{2505}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{27683}{37575}+\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\frac{3047}{7515}+\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{53248}{37575}-\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{2505}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{27683}{37575}-\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \left(\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-1\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{53248}{37575}+\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{2505}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{26977}{12525}-\frac{1121 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1711}{270}+\frac{128971 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{243391}{75150}-\frac{1219 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1350}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{27683}{37575}+\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{27683}{37575}-\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{243391}{75150}-\frac{1219 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{1350}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{345791}{225450}+\frac{11087 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)}{25050}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}-\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{2505}-\frac{847}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{158051 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}-\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{2505}-\frac{62521}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\frac{35141}{75150}-\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{7515}-\frac{62521 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{158051 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}-\frac{656 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{2505}-\frac{62521}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{158051 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{75150}+\frac{89413}{225450}-\frac{25687 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{12525}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{188}{7515}+\frac{89413 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{225450}-\frac{62521 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(\frac{35141}{75150}-\frac{847 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{7515}-\frac{62521 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{188}{7515}+\frac{89413 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{225450}-\frac{62521 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{75150}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)-\frac{21611}{45090}+\frac{35141 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)^{2}}{75150}-\frac{188 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{3047}{7515}+\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{6387}{4175}+\frac{3047 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{3047}{7515}-\mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{6387}{4175}-\frac{3047 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{7 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{2}-\frac{21329}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{44709}{8350}-\frac{21329 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{15030}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{206321 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{75150}+\frac{19063}{9018}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{97529}{15030}+\frac{19063 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{9018}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)+\left(-\frac{4961}{7515}-\frac{57319 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{112725}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)-\frac{204737}{112725}-\frac{4961 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{53402 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{37575}+\frac{638}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{862}{7515}+\frac{638 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =2\right)+\left(\frac{862}{7515}+\frac{638 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{1503}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =4\right)+\frac{20518}{37575}+\frac{862 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =3\right)}{7515}\right) \mathit{RootOf} \left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n}-\frac{3242 \mathit{RootOf}\left(6 Z^{5}-6 Z^{4}-21 Z^{3}+25 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n}}{37575}\right)}{2627641}\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 60 rules.
Finding the specification took 140 seconds.
Copy 60 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= x^{2}\\
F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{57}\! \left(x \right)\\
\end{align*}\)