Av(12453, 12543, 13452, 13542, 14352, 14532, 15342, 15432, 21453, 21543)
Counting Sequence
1, 1, 2, 6, 24, 110, 542, 2800, 14966, 82074, 459208, 2610938, 15042218, 87621664, 515190026, ...
Implicit Equation for the Generating Function
\(\displaystyle -F \left(x
\right)^{3}+\left(x +2\right) F \left(x
\right)^{2}-3 x F \! \left(x \right)+x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a{\left(n + 6 \right)} = \frac{5 n \left(n - 1\right) a{\left(n \right)}}{7 \left(n + 5\right) \left(n + 6\right)} - \frac{3 n \left(15 n + 11\right) a{\left(n + 1 \right)}}{7 \left(n + 5\right) \left(n + 6\right)} + \frac{6 \left(n + 1\right) \left(71 n + 120\right) a{\left(n + 2 \right)}}{35 \left(n + 5\right) \left(n + 6\right)} + \frac{3 \left(11 n + 52\right) a{\left(n + 5 \right)}}{5 \left(n + 6\right)} + \frac{6 \left(3 n^{2} - 55 n - 254\right) a{\left(n + 4 \right)}}{35 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(197 n^{2} + 301 n - 606\right) a{\left(n + 3 \right)}}{35 \left(n + 5\right) \left(n + 6\right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 110\)
\(\displaystyle a{\left(n + 6 \right)} = \frac{5 n \left(n - 1\right) a{\left(n \right)}}{7 \left(n + 5\right) \left(n + 6\right)} - \frac{3 n \left(15 n + 11\right) a{\left(n + 1 \right)}}{7 \left(n + 5\right) \left(n + 6\right)} + \frac{6 \left(n + 1\right) \left(71 n + 120\right) a{\left(n + 2 \right)}}{35 \left(n + 5\right) \left(n + 6\right)} + \frac{3 \left(11 n + 52\right) a{\left(n + 5 \right)}}{5 \left(n + 6\right)} + \frac{6 \left(3 n^{2} - 55 n - 254\right) a{\left(n + 4 \right)}}{35 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(197 n^{2} + 301 n - 606\right) a{\left(n + 3 \right)}}{35 \left(n + 5\right) \left(n + 6\right)}, \quad n \geq 6\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 44 rules.
Finding the specification took 2009 seconds.
Copy 44 equations to clipboard:
\(\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_{1}\! \left(x \right)+F_{41}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{30}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= y x\\
F_{19}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= -\frac{-F_{8}\! \left(x , y\right) y +F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= -\frac{-F_{12}\! \left(x , y\right) y +F_{12}\! \left(x , 1\right)}{-1+y}\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= -\frac{y \left(F_{14}\! \left(x , 1\right)-F_{14}\! \left(x , y\right)\right)}{-1+y}\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x , 1\right)\\
F_{43}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 92 rules.
Finding the specification took 2892 seconds.
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Copy 92 equations to clipboard:
\(\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_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= -F_{91}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= -F_{90}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{10}\! \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_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{28}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{62}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= -F_{84}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{0}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= \frac{F_{49}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{50}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{18}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{60}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{63}\! \left(x \right) &= \frac{F_{64}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{18}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{69}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= \frac{F_{71}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{71}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{18}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{62}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{0}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{32}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{0}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 92 rules.
Finding the specification took 7726 seconds.
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Copy 92 equations to clipboard:
\(\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_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= -F_{91}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= -F_{90}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{10}\! \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_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{61}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= -F_{84}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{18}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{0}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= -F_{54}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{18}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{60}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{68}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= \frac{F_{70}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{18}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{61}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{0}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{31}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{0}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right)\\
\end{align*}\)