Av(1342, 1432, 4213)
Generating Function
\(\displaystyle \frac{x^{5}-3 x^{3}+4 x^{2}-4 x +1}{x^{5}+x^{4}-6 x^{3}+7 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 256, 880, 3025, 10406, 35805, 123197, 423881, 1458425, 5017929, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+x^{4}-6 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-x^{5}+3 x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(4+n \right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(4+n \right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(10609 \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-\frac{85}{103}\right) \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{85}{103}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7855\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{142319}\\+\\\frac{47674 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{142319}\\-\\\frac{19364 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{142319}\\-\\\frac{10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-63654 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+53160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-15980\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-1854 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+630\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{142319}\\-\\\frac{21103 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{142319}\\+\\\frac{24576 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-63654 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+74263 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-28469\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-1854 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-1854 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+72409 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-52530\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-52530 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+32816\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{142319}\\+\\\frac{\left(\left(\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-6901 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-1530\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-1530 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(-8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-1530 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}+7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-10534\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{142319}\\+\\\frac{\left(\left(\left(\left(-10609 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+8755\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\left(8755 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-7225 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-7855 \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-18389\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-6 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{142319} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 105 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \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_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{58}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{95}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 3 F_{35}\! \left(x \right)+F_{101}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{57}\! \left(x \right)\\
\end{align*}\)