Av(1234, 1342, 1423, 1432, 2143)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{4}+3 x^{3}+3 x^{2}+x -1\right)}{x^{5}+2 x^{4}+x^{3}-x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 185, 581, 1825, 5734, 18017, 56613, 177891, 558976, 1756438, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+2 x^{4}+x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{4}+3 x^{3}+3 x^{2}+x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+3 a \! \left(4+n \right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+3 a \! \left(4+n \right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{5880 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{42967}\\-\\\frac{3086 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{42967}\\-\\\frac{5744 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{42967}\\-\\\frac{1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{42967}\\+\\\frac{\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+3598 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+4081\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{42967}\\+\\\frac{\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+3598 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-1287\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{42967}\\+\\\frac{\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{42967}\\+\\\frac{\left(\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-1452 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+4292\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+4292 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+6227\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{42967}\\+\\\frac{\left(\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{42967}\\+\\\frac{\left(\left(\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-3005 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+217\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{42967}\\+\\\frac{6894 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{42967}\\+\\\frac{\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{4}+3598 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+1497\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{42967}\\+\\\frac{\left(\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{3}+\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}-1452 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+4292\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}-1452 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+2493 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{3}+4292 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-649\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{42967}\\+\\\frac{\left(\left(\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{2}+\left(\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}-694 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-7297\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}-7297 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-6010\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{2}+\left(-2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}-7297 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-6010\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)-3005 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{2}-6010 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-3654\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{42967}\\+\\\frac{\left(\left(\left(\left(-1799 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+2146\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+3005 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-217\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)+\left(\left(2146 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)+3005\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)+3005 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-217\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)+\left(3005 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-217\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)-217 \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)-4088\right) \mathit{RootOf}\left(Z^{5}+2 Z^{4}+Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{42967} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 97 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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x^{2}\\
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_{90}\! \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_{35}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= x^{2}\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \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_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{78}\! \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_{44}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{46}\! \left(x \right)+F_{72}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \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_{4}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \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_{62}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{72}\! \left(x \right)+F_{80}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{71}\! \left(x \right)\\
\end{align*}\)