Av(1243, 1324, 1432, 2134, 3214)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{2 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 160, 456, 1305, 3743, 10739, 30805, 88354, 253411, 726828, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}+x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{1962944742 \left(\left(\left(\left(-1+\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2563}{13358}+\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{4688}{6679}-\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{20439}{146938}-\frac{19253 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{145995}{146938}-\frac{60979 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{81156}{73469}+\frac{24214 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{2563}{13358}+\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{145995}{146938}-\frac{60979 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{4548465}{587752}+\frac{3208307 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{1130343 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{146938}+\frac{1860301}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{4688}{6679}-\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{81156}{73469}+\frac{24214 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1130343 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{146938}+\frac{1860301}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1749557 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{293876}-\frac{3351039}{587752}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{54708}{73469}-\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}-\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{35725}{73469}+\frac{120381 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{611}{13358}-\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{35725}{73469}+\frac{120381 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{645127}{73469}+\frac{3370057 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}+\frac{120381 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{645127 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{2072183}{293876}-\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{611}{13358}-\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{645127 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{2072183}{293876}-\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{2072183 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}-\frac{755423}{146938}-\frac{611 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+1\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{81223}{146938}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{34904 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}-\frac{3877}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{297559}{146938}-\frac{3877 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{536249 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{962851}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1651017}{587752}-\frac{962851 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{75624 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6679}-\frac{633245}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{15425}{13358}-\frac{633245 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{2368373}{293876}-\frac{1510159 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{2368373 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}-\frac{2022111}{587752}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2563}{13358}+\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{4688}{6679}-\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{2563}{13358}+\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{120381 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{146938}-\frac{327169}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{117465}{146938}+\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{4688}{6679}-\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{117465}{146938}+\frac{35725 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{611 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13358}-\frac{329347}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+1\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{81223}{146938}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{34904 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}-\frac{3877}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{297559}{146938}-\frac{3877 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1577 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{35179}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{496953}{587752}-\frac{35179 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{4077}{13358}+\frac{11511 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{4077 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13358}-\frac{440091}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+1\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{81223}{146938}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{15651 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{2563}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{470069}{293876}+\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{4688}{6679}+\frac{54708 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6679}-\frac{55372}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{1360791 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}-\frac{2097911}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{3018157}{293876}-\frac{2097911 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{3018157}{293876}-\frac{2097911 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{3018157 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}-\frac{7432839}{587752}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{267336 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{231918}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{144258}{73469}+\frac{231918 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{144258}{73469}+\frac{231918 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{144258 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{397233}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{50555 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{35947}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{125049}{293876}+\frac{35947 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{125049}{293876}+\frac{35947 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{125049 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}-\frac{1119317}{587752}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{77130 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}-1\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{81223}{146938}-\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{81223}{146938}-\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{81223 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{676341}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{81223}{146938}-\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{470069}{293876}-\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{55372}{73469}+\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-\left(\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1}{2}\right) \left(\left(-\frac{81223}{146938}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{470069}{293876}+\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{55372}{73469}-\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{470069}{293876}-\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{2463}{53432}+\frac{16189 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{26716}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3544573}{293876}-\frac{1009851 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{368540 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}-\frac{581972}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{55372}{73469}+\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{27686}{73469}+\frac{2344 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{368540 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}-\frac{581972}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{301794 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{73469}+\frac{513580}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{4688}{6679}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}+\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{117465}{146938}+\frac{327169 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}+\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{329347}{293876}-\frac{117465 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{117465}{146938}+\frac{327169 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}+\frac{2563 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{13358}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{843021}{146938}-\frac{422905 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{53432}+\frac{327169 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{843021 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{2085005}{587752}-\frac{117465 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{329347}{293876}-\frac{117465 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{4688 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6679}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{843021 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}-\frac{2085005}{587752}-\frac{117465 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{2085005 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{587752}+\frac{978845}{293876}+\frac{329347 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{81223}{146938}+\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{81223 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}+\frac{676341}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{\mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2}-\frac{81223}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{676341}{587752}-\frac{81223 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-3 \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)+\frac{243669}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{2029023}{293876}+\frac{243669 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{572999 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{1011811}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{782763}{293876}+\frac{1011811 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{415856}{73469}+\frac{523244 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{415856 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{190526}{73469}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{866875 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{73469}+\frac{1336703}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1922601}{293876}+\frac{1336703 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{1922601}{293876}+\frac{1336703 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{146938}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{1922601 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{293876}+\frac{4743467}{587752}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}+\frac{91363 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{73469}\right) \left(\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{8942}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{8942 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}+\frac{7661}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{8942 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}+\frac{7661}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{7661 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}-\frac{5552}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{8942 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}+\frac{7661}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{7661 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}-\frac{5552}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{7661 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}-\frac{5552}{13359}\right) \mathit{RootOf} \left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{5552 \mathit{RootOf}\left(2 Z^{5}+Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13359}-\frac{5195}{13359}\right)}{8347197769}\)
This specification was found using the strategy pack "Point Placements" and has 83 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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{64}\! \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_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{70}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= x^{2}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
\end{align*}\)