Av(1342)
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Generating Function
\(\displaystyle \frac{-8 \sqrt{-8 x +1}\, x -8 x^{2}+\sqrt{-8 x +1}+20 x +1}{2 \left(x +1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right)^{3} F \left(x \right)^{2}+\left(8 x^{2}-20 x -1\right) F \! \left(x \right)+16 x = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +2}+\frac{\left(-8+7 n \right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)
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This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 29 rules.

Found on May 26, 2021.

Finding the specification took 1720 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_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)^{2} F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{4}\! \left(x \right) F_{6}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{17}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= \frac{y F_{24}\! \left(x , y\right)-F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{28}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{11}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 32 rules.

Found on May 26, 2021.

Finding the specification took 3304 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_{27}\! \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_{27}\! \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_{14}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)^{2} F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{16}\! \left(x , y\right) F_{27}\! \left(x \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)+F_{31}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= x\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{27}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\ F_{31}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{10}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion Req Corrob" and has 46 rules.

Found on May 26, 2021.

Finding the specification took 1809 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_{13}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{0}\! \left(x \right) F_{13}\! \left(x \right)}\\ F_{12}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{45}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)^{2} F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= y x\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{24}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{24}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= -\frac{y \left(F_{15}\! \left(x , 1\right)-F_{15}\! \left(x , y\right)\right)}{-1+y}\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{24}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{30}\! \left(x , y\right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Req Corrob" and has 79 rules.

Found on May 26, 2021.

Finding the specification took 10962 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)^{2} F_{14}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{20}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{8}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= -\frac{F_{12}\! \left(x , 1\right)-F_{12}\! \left(x , y\right)}{-1+y}\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= -\frac{y \left(F_{12}\! \left(x , 1\right)-F_{12}\! \left(x , y\right)\right)}{-1+y}\\ F_{44}\! \left(x , y\right) &= -\frac{-y F_{45}\! \left(x , y\right)+F_{45}\! \left(x , 1\right)}{-1+y}\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x \right)+F_{59}\! \left(x , y\right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x , 1\right)\\ F_{47}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{51}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{55}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{52}\! \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_{50}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{69}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{52}\! \left(x \right)+F_{72}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{75}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{76}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion Req Corrob" and has 94 rules.

Found on May 26, 2021.

Finding the specification took 1798 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_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{24}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)^{2} F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= y x\\ F_{25}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{2}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{62}\! \left(x , y\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_{2}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{6}\! \left(x \right)\\ 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_{8}\! \left(x \right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= -\frac{F_{36}\! \left(x , 1\right) y -F_{36}\! \left(x , y\right)}{-1+y}\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= -\frac{y \left(F_{17}\! \left(x , 1\right)-F_{17}\! \left(x , y\right)\right)}{-1+y}\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= -\frac{F_{44}\! \left(x , 1\right) y -F_{44}\! \left(x , y\right)}{-1+y}\\ F_{44}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= -\frac{y \left(F_{30}\! \left(x , 1\right)-F_{30}\! \left(x , y\right)\right)}{-1+y}\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= -\frac{F_{48}\! \left(x , 1\right) y -F_{48}\! \left(x , y\right)}{-1+y}\\ F_{48}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{36}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= -\frac{y \left(F_{50}\! \left(x , 1\right)-F_{50}\! \left(x , y\right)\right)}{-1+y}\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{54}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{55}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{56}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{60}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{58}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{69}\! \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_{8}\! \left(x \right)}\\ F_{71}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{2}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{74}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{73}\! \left(x \right)\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{8}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\ \end{align*}\)