Av(1324, 1342, 1432, 3214)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{4}}{\left(3 x -1\right) \left(x^{5}-2 x^{4}+6 x^{3}-7 x^{2}+4 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 201, 603, 1793, 5343, 15990, 47976, 144041, 432355, 1297280, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -1\right) \left(x^{5}-2 x^{4}+6 x^{3}-7 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+7 a \! \left(n +1\right)-20 a \! \left(n +2\right)+27 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+7 a \! \left(n +1\right)-20 a \! \left(n +2\right)+27 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{8233119560 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{10979}{13290}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{10979 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{13290}+\frac{895}{1329}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{10979 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{13290}+\frac{895}{1329}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{895 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{1329}-\frac{19753}{26580}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{10979 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{13290}+\frac{895}{1329}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{895 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{1329}-\frac{19753}{26580}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{895 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{1329}-\frac{19753}{26580}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{19753 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{26580}-\frac{25787}{26580}\right) \left(\left(\left(\left(1-\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{630800}{464623}+\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{208215}{464623}-\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(1-\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{1827033}{464623}+\frac{3032961 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{72205}{20201}-\frac{3107773 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{17145}{20201}+\frac{935841 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{630800}{464623}+\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{72205}{20201}-\frac{3107773 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{5401680}{464623}+\frac{9704718 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{2796872}{464623}-\frac{5041975 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{208215}{464623}-\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{17145}{20201}+\frac{935841 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2796872}{464623}-\frac{5041975 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{2841062 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}-\frac{1680730}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}+\frac{363813}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}+\frac{4352989 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{1871481}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{871059}{929246}-\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}+\frac{4352989 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{1871481}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{4352989 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}-\frac{2321530}{464623}+\frac{4707215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{3640561}{929246}-\frac{2321530 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{871059}{929246}-\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3640561}{929246}-\frac{2321530 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{3640561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{1381285}{464623}+\frac{871059 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{266987}{464623}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{1976381 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{1196233}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{516659}{464623}+\frac{1196233 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{5464520 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{3321712}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{1567292}{464623}-\frac{3321712 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1167797 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{840745}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{1695662}{464623}-\frac{840745 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{1591772}{464623}-\frac{2442667 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{1813300}{464623}+\frac{1591772 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(1-\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{630800}{464623}+\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{208215}{464623}-\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{630800}{464623}+\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4352989 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}+\frac{2253281}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{826869}{929246}+\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{208215}{464623}-\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{826869}{929246}+\frac{1871481 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{299445}{929246}-\frac{871059 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{266987}{464623}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{1976381 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{1196233}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{516659}{464623}+\frac{1196233 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1862557 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{1068149}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{107459}{929246}-\frac{1068149 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{38199}{929246}-\frac{201 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{387825}{929246}-\frac{38199 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{266987}{464623}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1056580 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{630800}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{8300}{20201}+\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{208215}{464623}+\frac{363813 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{22095}{464623}-\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{9246688 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{5695028}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{3460681}{464623}-\frac{5695028 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{3460681}{464623}-\frac{5695028 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{1692095}{464623}+\frac{3460681 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{9066483 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{5575275}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{3242043}{929246}+\frac{5575275 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{3242043}{929246}+\frac{5575275 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{180045}{929246}-\frac{3242043 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{919801 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\frac{565433}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{325759}{464623}-\frac{565433 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{325759}{464623}-\frac{565433 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{545}{20201}+\frac{325759 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-\frac{755879 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{266987}{464623}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{266987}{464623}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{17315}{464623}-\frac{266987 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{266987}{464623}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{8300}{20201}-\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{22095}{464623}+\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-2\right) \left(\left(-\frac{266987}{464623}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{8300}{20201}-\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{22095}{464623}+\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8300}{20201}-\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{359705}{464623}+\frac{63905 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{20201}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3447199}{929246}-\frac{9982511 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1609085}{929246}+\frac{5275951 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{22095}{464623}+\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{44190}{464623}-\frac{416430 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1609085}{929246}+\frac{5275951 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{3062645 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)}{929246}+\frac{1443115}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}-\frac{208215}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}-\frac{2253281 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{826869}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{299445}{929246}+\frac{826869 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{630800 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}-\frac{2253281 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{826869}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{2253281 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}+\frac{1661384}{464623}-\frac{3822211 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{2463755}{929246}+\frac{1661384 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{826869 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{299445}{929246}+\frac{826869 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}-\frac{208215 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{2463755}{929246}+\frac{1661384 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{826869 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{2463755 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{2287417}{929246}-\frac{299445 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{266987}{464623}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{17315}{464623}+\frac{266987 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(2 \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{533974}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{34630}{464623}-\frac{533974 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-6 \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{1601922}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{103890}{464623}+\frac{1601922 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1580051 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{1244779}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{3323285}{929246}+\frac{1244779 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{1874225}{929246}+\frac{2777371 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2690539}{929246}-\frac{1874225 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{8084773 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}+\frac{4982597}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{3080875}{929246}+\frac{4982597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{3080875}{929246}+\frac{4982597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2018045}{929246}-\frac{3080875 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{929246}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}+\frac{682689 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{929246}+\left(\left(\left(-\frac{5152952 \,3^{n}}{464623}+\frac{8361976 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{5152952 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{3178168 \,3^{n}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{5152952 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{3178168 \,3^{n}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2005640 \,3^{n}}{464623}+\frac{3178168 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{5152952 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}+\frac{3178168 \,3^{n}}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{2005640 \,3^{n}}{464623}+\frac{3178168 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{2005640 \,3^{n}}{464623}+\frac{3178168 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{1697080 \,3^{n}}{464623}-\frac{2005640 \,3^{n} \mathit{RootOf}\left(Z^{5}-2 Z^{4}+6 Z^{3}-7 Z^{2}+4 Z -1, \mathit{index} =3\right)}{464623}\right)}{51784918969}\)
This specification was found using the strategy pack "Point Placements" and has 68 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 68 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_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{45}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{44}\! \left(x \right)\\
\end{align*}\)