Av(1243, 1342, 2143, 2314, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{5}-3 x^{4}-x^{3}+4 x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 159, 447, 1259, 3556, 10056, 28441, 80426, 227405, 642969, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{4}-x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{1816227 \left(\left(\left(\left(\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{5938}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{4331}{2043}+\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{9874 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{13268}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{292 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{2289}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{1960}{227}-\frac{9880 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{5938}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{292 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{2289}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{10986}{227}-\frac{9266 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{9085 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}-\frac{10745}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{4331}{2043}+\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1960}{227}-\frac{9880 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{9085 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}-\frac{10745}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{23042}{2043}-\frac{534 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{1579}{681}-\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1574}{2043}-\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{7771 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{2371}{2043}-\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{681}-\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1574}{2043}-\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{7771 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{48424}{2043}-\frac{43291 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{7771 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{48424 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}+\frac{3814}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{2371}{2043}-\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{681}-\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{48424 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}+\frac{3814}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{24157}{2043}+\frac{2371 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}+\frac{3814 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1201}{2043}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{7330}{2043}+\frac{4445 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{26150}{2043}-\frac{7330 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{520}{681}-\frac{2365 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{90460}{2043}+\frac{520 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{8324}{2043}+\frac{373 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{29278}{2043}-\frac{8324 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{475}{2043}-\frac{3175 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{475 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\frac{53675}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-1\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{5938}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{4331}{2043}+\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{5938}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3962}{227}+\frac{7771 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{28262}{2043}+\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{4331}{2043}+\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{28262}{2043}+\frac{1574 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{2371 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-\frac{15733}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1201}{2043}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{7330}{2043}+\frac{4445 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{26150}{2043}-\frac{7330 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1673}{227}+\frac{831 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{99881}{2043}+\frac{1673 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{10622}{2043}+\frac{3818 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{10622 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\frac{76166}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1201}{2043}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{5938}{2043}-\frac{3461 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{34084}{2043}-\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{4331}{2043}-\frac{1579 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{10211}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{5899}{681}+\frac{20456 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{481}{227}-\frac{5899 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{481}{227}-\frac{5899 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{481 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}+\frac{172687}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{4499}{681}+\frac{9844 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{9421}{2043}+\frac{4499 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{9421}{2043}+\frac{4499 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{9421 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\frac{5157}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{464}{681}-\frac{16796 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{7934}{2043}+\frac{464 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{7934}{2043}+\frac{464 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{7934 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{39964}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{3764 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1201}{2043}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{1201}{2043}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1201 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\frac{29753}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{1201}{2043}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{34084}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{10211}{681}+\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1201}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-\frac{34084}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{10211}{681}-\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-3\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}+\frac{34084}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{13483 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}-\frac{23873}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{163544}{2043}-\frac{7487 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{13832 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}-\frac{67103}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{10211}{681}+\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{10211}{227}-\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{13832 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{681}-\frac{67103}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{76967}{2043}+\frac{5306 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{4331}{2043}-\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{28262}{2043}-\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{3962 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{15733}{2043}-\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{28262 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{28262}{2043}-\frac{5938 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{3962 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{59023}{2043}-\frac{16933 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{3962 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{227}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\frac{59023 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{28262 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{41893}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\frac{15733}{2043}-\frac{4331 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{28262 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{59023 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{28262 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{41893}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)-\frac{116116}{2043}+\frac{15733 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{2}}{2043}-\frac{41893 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{1201}{2043}+\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{1201 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{29753}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{1201}{681}-3 \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{29753}{681}+\frac{1201 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{1201}{2043}-\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{1201 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}+\frac{29753}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{37561}{2043}-\frac{48452 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{158243}{2043}-\frac{37561 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(-\frac{36470}{2043}-\frac{37165 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{36470 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}-\frac{221560}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{10919}{681}-\frac{56624 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{4359}{227}-\frac{10919 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{4359}{227}-\frac{10919 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{681}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{4359 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{227}-\frac{341773}{2043}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}-\frac{118190 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{2043}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)+\frac{785}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{785 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}-\frac{397}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(\frac{785 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}-\frac{397}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{397 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}+\frac{261}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)+\left(\left(\frac{785 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}-\frac{397}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)-\frac{397 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}+\frac{261}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)+\left(-\frac{397 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}+\frac{261}{889}\right) \mathit{RootOf} \left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)+\frac{261 \mathit{RootOf}\left(Z^{5}-3 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)}{889}-\frac{3625}{889}\right)}{2793775220}\)
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This specification was found using the strategy pack "Point Placements" and has 45 rules.

Found on January 18, 2022.

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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_{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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \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_{41}\! \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_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \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_{7}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \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_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{34}\! \left(x \right)\\ \end{align*}\)