Av(1243, 1342, 3124, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{5}+3 x^{3}-4 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 63, 191, 574, 1727, 5205, 15697, 47340, 142761, 430502, 1298189, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+4 a \! \left(n +3\right)-4 a \! \left(n +4\right)+a \! \left(n +5\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{129217109 \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{4021}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{4021 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}-\frac{3189}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{4021 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}-\frac{3189}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{3189 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}+\frac{2258}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{4021 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}-\frac{3189}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{3189 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}+\frac{2258}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{3189 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}+\frac{2258}{2947}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{2258 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{2947}-\frac{5048}{2947}\right) \left(\left(\left(\left(\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{374027}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{367994}{131541}+\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{136354 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{936209}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{185872 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}+\frac{739972}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1115888}{131541}+\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{374027}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{185872 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}+\frac{739972}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{158768 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{1017068}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{32940 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}+\frac{173744}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{367994}{131541}+\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{1115888}{131541}+\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{32940 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}+\frac{173744}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{82912}{43847}+\frac{173744 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{43847}+\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{188155}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{281639 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{170810}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}-\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{512060}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{281639 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{170810}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{583500 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{281639 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{190588}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{43847}+\frac{190588 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{793472}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}-\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{512060}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{43847}+\frac{190588 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{793472}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{512060 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{82912}{43847}+\frac{793472 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{187394 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{1328}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(3-\frac{167808 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+3 \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{562182}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{406967}{43847}-\frac{64929 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{406967 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{198234}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{155215 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{194250}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{194250 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{1198800}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{374027}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{367994}{131541}+\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{374027}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{884408}{131541}-\frac{281639 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-\frac{201276}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{367994}{131541}+\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{170810 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-\frac{201276}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{512060 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{272240}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{187394 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{1328}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{144436}{131541}-\frac{95767 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{144436 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{512060}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{144436 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{512060}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{512060 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{272240}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{374027}{131541}+\frac{4813 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{371978}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{367994}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{1115888}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{20173}{43847}-\frac{2578 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{269}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{20173 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{555326}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{20173 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{555326}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{555326 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{1193488}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{250187}{131541}+\frac{407657 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{250187 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{1174486}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{250187 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{1174486}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{1174486 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{260288}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{188155}{131541}-\frac{126728 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{367994}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\frac{188155 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{367994}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{1115888}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{55936 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)-\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{187394 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{1328}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}+\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}-\frac{367994}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}-\frac{884408 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{201276}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{201276 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{272240}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}-\frac{884408 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{201276}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{112891 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{884408 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{2607431}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{201276 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{43847}+\frac{2607431 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{2313991}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}+\frac{201276 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{272240}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{201276 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{43847}+\frac{2607431 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}+\frac{2313991}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{272240 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}}{131541}-\frac{689794}{43847}+\frac{2313991 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}+\frac{371978}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{1115888}{131541}+\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}-\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right) \left(\left(\frac{187394}{43847}+\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}-\frac{371978}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{1115888}{131541}-\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{374027 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}+\frac{371978}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{1115888}{131541}+\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{515443 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}-\frac{86647}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\frac{778397 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}+\frac{439217}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-\frac{1115888}{131541}+\frac{367994 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{778397 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{43847}+\frac{439217}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)-\frac{355885}{131541}+\frac{2313991 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)}{131541}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{187394}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{187394 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{1328}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+\left(\left(3 \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)+\frac{562182}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{562182 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{3984}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{458625}{43847}-\frac{141416 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)-\frac{458625 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{766994}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(\frac{410403 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{1555105}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{1555105 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{1053102}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{290951}{43847}+\frac{33972 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{290951 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{761682}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\frac{290951 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}+\frac{761682}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)+\frac{761682 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)}{43847}-\frac{1235184}{43847}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}-\frac{1263457 \mathit{RootOf}\left(Z^{5}+3 Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{43847}\right)}{93901387697}\)
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This specification was found using the strategy pack "Point Placements" and has 61 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{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_{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_{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_{44}\! \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_{28}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \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_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \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_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{33}\! \left(x \right)\\ \end{align*}\)