Av(1342, 1432, 2143, 3124, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}+x -1\right)}{x^{5}+3 x^{4}-x^{2}+3 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 153, 424, 1182, 3306, 9250, 25869, 72327, 202212, 565365, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{4}-x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+\frac{a \! \left(n +3\right)}{3}-a \! \left(n +4\right)+\frac{a \! \left(n +5\right)}{3}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{13473873 \left(\left(\left(\left(-\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{55923}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{136276 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{190864}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{187130 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{205805}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{56072 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{48243}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{55923}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{187130 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{205805}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{338608 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{259287}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{205830 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{187695}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{56072 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{48243}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{205830 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{187695}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{206307 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{242108}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}+\frac{12641}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}+\frac{71644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{22839}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\frac{7105}{30553}+\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}+\frac{71644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{22839}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{35272 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{129307}{30553}+\frac{71644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{129307 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{197942}{30553}+\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(\frac{7105}{30553}+\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{129307 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{197942}{30553}+\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\frac{7105 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}-\frac{197942 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{280958}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1+\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\frac{134941}{30553}+\frac{96754 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{134941 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{131584}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{129846}{30553}+3 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{129846 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{5214}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{151383}{30553}-\frac{114767 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{151383 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{206307}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{151383 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{206307}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{206307 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{242108}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{55923}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{55923}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{42726}{30553}-\frac{71644 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{901}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{22839 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{901}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{7105 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}+\frac{12950}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1+\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{134941}{30553}+\frac{96754 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{134941 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{131584}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{163079}{30553}+\frac{115486 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{163079 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{52556}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(\frac{33233 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{47342}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{47342 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{5662}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1+\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{55923}{30553}+\frac{39522 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{19887}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(\frac{12641 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{6204}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{120830}{30553}-\frac{92700 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{120830 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{163025}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(-\frac{120830 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{163025}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{163025 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{240370}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{33233}{30553}+\frac{23827 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{33233 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{47342}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(\frac{33233 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{47342}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{47342 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{5662}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{79018}{30553}-\frac{57232 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{79018 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{111697}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(-\frac{79018 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{111697}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{111697 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{11418}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1-\frac{22067 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)-\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)-\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{43282 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{1738}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{19887}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}+\frac{6204}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}-\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+3\right) \left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}+\frac{19887}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{6204}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{19887}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{185918 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{53457}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{358703 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{393677}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{255281 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{351564}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}+\frac{6204}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{54447 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}+\frac{18612}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{255281 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{351564}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{286523 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{30553}-\frac{220632}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}+\frac{18149}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}+\frac{42726 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{901}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{12950}{30553}+\frac{901 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{55923 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}+\frac{42726 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{901}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{228723 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{265528}{30553}+\frac{42726 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{265528 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{325373}{30553}+\frac{901 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{12950}{30553}+\frac{901 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{18149 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{265528 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{325373}{30553}+\frac{901 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)-\frac{12950 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{2}}{30553}-\frac{325373 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{196377}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{43282}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{43282 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{1738}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\frac{129846}{30553}+3 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{129846 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}+\frac{5214}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{412289}{30553}-\frac{304256 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{412289 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{513701}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(-\frac{255281 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{351564}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{351564 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{281516}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{369007}{30553}-\frac{273703 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{369007 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{511963}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(-\frac{369007 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{511963}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)-\frac{511963 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{30553}-\frac{330012}{30553}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n}+\frac{223595 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n}}{30553}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{1186}{2205}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{1186 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{2205}+\frac{41}{105}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(\frac{1186 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{2205}+\frac{41}{105}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{41 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{105}+\frac{127}{315}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+\left(\left(\frac{1186 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{2205}+\frac{41}{105}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{41 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{105}+\frac{127}{315}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+\left(\frac{41 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{105}+\frac{127}{315}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{127 \mathit{RootOf}\left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{315}+\frac{1856}{441}\right)}{1999788961}\)
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This specification was found using the strategy pack "Point Placements" and has 42 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_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{27}\! \left(x \right)\\ \end{align*}\)