Av(1234, 1342, 1432, 2143, 2413)
Generating Function
\(\displaystyle -\frac{x^{3}+2 x -1}{2 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 155, 451, 1313, 3814, 11078, 32187, 93522, 271723, 789471, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)+x^{3}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{3657490415 \left(\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{15741}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{15741 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{1626}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\frac{15741 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{1626}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1626 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{13029}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{15741 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{1626}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1626 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{13029}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{1626 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}+\frac{13029}{12461}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{13029 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{12461}-\frac{69593}{12461}\right) \left(\left(\left(\left(-\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{293515}-1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}+\frac{158209}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{184533}{117406}-\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{293515}-1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{74813 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{293515}+\frac{232085}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{273351 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{1174060}+\frac{4302731}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{8165601}{1174060}-\frac{280017 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}+\frac{158209}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{273351 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{1174060}+\frac{4302731}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1779397 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{1174060}-\frac{4403889}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{696991}{587030}+\frac{815393 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{184533}{117406}-\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{8165601}{1174060}-\frac{280017 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{696991}{587030}+\frac{815393 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{1187623 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{234812}-\frac{14773207}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{89049}{117406}-\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}+\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{293515}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1887}{58703}-\frac{96791 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}+\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}+\frac{73317}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1887}{58703}-\frac{96791 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2127123 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+\frac{9344833}{587030}-\frac{96791 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{58703}-\frac{2895913}{587030}+\frac{9344833 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}+\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}+\frac{73317}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{58703}-\frac{2895913}{587030}+\frac{9344833 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{2895913 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}+\frac{73317 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}-\frac{3757411}{293515}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{268623 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{305961}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{305961 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}-\frac{3774073}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{930329 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}+\frac{6223}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{6223 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}+\frac{3432399}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1248231 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{1174060}+\frac{1762671}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1762671 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}+\frac{1000991}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{1268155 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{1556689}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{23861473}{1174060}-\frac{1556689 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{293515}-1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}+\frac{158209}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{184533}{117406}-\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}+\frac{158209}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{96791 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}+\frac{3450683}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{58703}-\frac{2455359}{293515}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{184533}{117406}-\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1887 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{58703}-\frac{2455359}{293515}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{256437}{587030}-\frac{73317 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{268623 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{305961}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{305961 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}-\frac{3774073}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{79769 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{1174060}+\frac{519727}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{519727 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}+\frac{2736383}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{272469 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{331167}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{5057007}{1174060}-\frac{331167 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{418249 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{158209}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}-\frac{3469553}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{89049 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}+\frac{184533}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{4544133}{587030}+\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{1785477 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{1174060}+\frac{1150749}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1150749 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{1309431}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{1150749 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{1309431}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{3893851}{234812}-\frac{1309431 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{1780889 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{1174060}+\frac{507281}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{507281 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{825683}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{507281 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}-\frac{825683}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{13841443}{1174060}-\frac{825683 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{343436 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+\frac{73876}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{73876 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}+\frac{30452}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\frac{73876 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}+\frac{30452}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{1174012}{293515}+\frac{30452 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-\frac{318407 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}-1\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{2196109}{293515}+\frac{123629 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}+\frac{3469553}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{4544133}{587030}-\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-\left(-\frac{1}{2}+\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)\right) \left(\left(\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}-\frac{3469553}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{4544133}{587030}+\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}+\frac{3469553}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{527275 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{234812}-\frac{12557819}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{9608849 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{1174060}-\frac{8108083}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{1733083}{587030}-\frac{2514697 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{4544133}{587030}-\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{4544133}{1174060}+\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{234812}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1733083}{587030}-\frac{2514697 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{6875221 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)}{1174060}-\frac{36777217}{1174060}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{184533}{117406}-\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}+\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2455359}{293515}-\frac{3450683 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}+\frac{2455359 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+\frac{256437}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2455359}{293515}-\frac{3450683 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{158209 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8175601 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}-\frac{4713619}{587030}-\frac{3450683 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\frac{2455359 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{293515}-\frac{3565829}{587030}-\frac{4713619 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(\frac{184533 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{117406}+\frac{2455359 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}+\frac{256437}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2455359 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{293515}-\frac{3565829}{587030}-\frac{4713619 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)-\frac{3565829 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}+\frac{256437 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}}{587030}-\frac{1759715}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)-\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{2196109}{293515}-\frac{123629 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{\mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{2}+\frac{123629}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{123629 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}+\frac{2196109}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)+\frac{123629}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)+\frac{2196109}{293515}+\frac{123629 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{5134137 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{587030}-\frac{571311}{117406}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{571311 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{117406}-\frac{6568079}{587030}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(-\frac{1718681 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}-\frac{281105}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{7254961}{293515}-\frac{281105 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{2420311 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{293515}-\frac{223841}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{223841 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}-\frac{437197}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-\frac{223841 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}-\frac{437197}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)-\frac{10858197}{293515}-\frac{437197 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)}{58703}\right) \mathit{RootOf} \left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}-\frac{10551536 \mathit{RootOf}\left(2 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{293515}\right)}{3479215998728}\)
This specification was found using the strategy pack "Point Placements" and has 94 rules.
Found on January 18, 2022.Finding the specification took 2 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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= x^{2}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{70}\! \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_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{64}\! \left(x \right)+F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{62}\! \left(x \right)\\
\end{align*}\)