Av(1243, 1342, 2143, 2314, 2413)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x -1\right)^{2}}{x^{5}+3 x^{4}-10 x^{3}+12 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 173, 512, 1513, 4471, 13213, 39047, 115385, 340950, 1007440, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{4}-10 x^{3}+12 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-3 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{109470400 \left(\left(\left(\left(-1+\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{832}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{37333}{20800}-\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}-\frac{539}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{832}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{2101}{2600}+\frac{11347 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{3399}{1600}-\frac{146527 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{9151 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2600}-\frac{291}{520}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{37333}{20800}-\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{3399}{1600}-\frac{146527 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1941}{416}+\frac{22317 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{43637}{20800}+\frac{16971 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}-\frac{539}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{9151 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2600}-\frac{291}{520}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{43637}{20800}+\frac{16971 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{47569 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}-\frac{4031}{2600}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{14537}{10400}-\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{832}+\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{22493}{20800}+\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{10400}-\frac{16049 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4160}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{10400}+\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}-\frac{791}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{22493}{20800}+\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{10400}-\frac{16049 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4160}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{64691}{20800}-\frac{16049 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4160}-\frac{9353 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2600}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{19387}{10400}+\frac{64691 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}+\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{10400}+\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}-\frac{791}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{19387}{10400}+\frac{64691 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}+\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{19387 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}+\frac{22613}{20800}-\frac{791 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}+1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{8259}{20800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{3293 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}+\frac{54141}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{15007}{20800}+\frac{54141 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{8123 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{416}-\frac{232777}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{7767}{800}-\frac{232777 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{850353 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}+\frac{224657}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{585549}{20800}+\frac{224657 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{187709}{20800}+\frac{91681 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{67943}{5200}-\frac{187709 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{832}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{37333}{20800}-\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}-\frac{539}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{37333}{20800}-\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{16049 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4160}-\frac{76291}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{1291}{800}-\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}-\frac{539}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1291}{800}-\frac{22493 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{18287}{20800}+\frac{791 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}+1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{8259}{20800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{3293 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}+\frac{54141}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{15007}{20800}+\frac{54141 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{4361 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{400}-\frac{60239}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{82279}{10400}-\frac{60239 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{22603}{10400}-\frac{95701 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{109493}{20800}+\frac{22603 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}+1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{8259}{20800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{35489 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}-\frac{37333}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{26899}{10400}-\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{539}{800}-\frac{14537 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{15201}{10400}+\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{17681 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{1600}+\frac{29281}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{88601}{20800}+\frac{29281 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{88601}{20800}+\frac{29281 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{51409}{5200}-\frac{88601 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{89689 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}+\frac{112299}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{4673}{2600}+\frac{112299 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{4673}{2600}+\frac{112299 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{288347}{20800}-\frac{4673 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2600}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{153303 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}-\frac{45737}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{38791}{20800}-\frac{45737 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{38791}{20800}-\frac{45737 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{74877}{10400}+\frac{38791 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{1375 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{832}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{8259}{20800}-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{8259}{20800}-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{4973}{2600}+\frac{8259 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{8259}{20800}-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{26899}{10400}+\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{800}+\frac{15201}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}-\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+3\right) \left(\left(-\frac{8259}{20800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{26899}{10400}-\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{800}-\frac{15201}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{26899}{10400}+\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{8187}{1300}+\frac{19597 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4160}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{6871}{320}-\frac{2459 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5200}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{55311}{5200}-\frac{649 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{800}+\frac{15201}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{1617 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{800}+\frac{45603}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{55311}{5200}-\frac{649 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{19437 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)}{10400}+\frac{66433}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{539}{800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}-\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1291}{800}-\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}+\frac{76291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{800}-\frac{1291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{800}+\frac{18287}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1291}{800}-\frac{37333 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}+\frac{76291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{116159}{20800}+\frac{76291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}+\frac{2337 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{320}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{18747}{4160}-\frac{116159 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}-\frac{1291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{539 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{800}-\frac{1291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{800}+\frac{18287}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{18747}{4160}-\frac{116159 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}-\frac{1291 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{18747 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4160}-\frac{3067}{1300}+\frac{18287 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{8259}{20800}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{4973}{2600}-\frac{8259 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(3 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{24777}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{14919}{2600}-\frac{24777 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-10 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{8259}{2080}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{4973}{260}+\frac{8259 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2080}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{12673 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{650}-\frac{182571}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{53143}{2080}-\frac{182571 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{10347}{2600}-\frac{209 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{25}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{44237}{5200}+\frac{10347 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2600}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{4873 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{650}-\frac{83463}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{27011}{10400}-\frac{83463 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{27011}{10400}-\frac{83463 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{20800}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{53497}{20800}+\frac{27011 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{10400}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}+\frac{13219 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{20800}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{3354}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{3354 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}+\frac{2474}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{3354 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}+\frac{2474}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{2474 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}-\frac{1549}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{3354 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}+\frac{2474}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{2474 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}-\frac{1549}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{2474 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}-\frac{1549}{5263}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{1549 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5263}-\frac{1301}{5263}\right)}{174741961}\)
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This specification was found using the strategy pack "Point Placements" and has 48 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 0\\ F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ \end{align*}\)