Av(2143, 2413, 264351)
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Generating Function
\(\displaystyle -\frac{x}{2}+\frac{3}{2}-\frac{\sqrt{x^{2}-6 x +1}}{2}\)
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Counting Sequence
1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, ...
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Implicit Equation for the Generating Function
\(\displaystyle F \left(x \right)^{2}+\left(x -3\right) F \! \left(x \right)+2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2\)
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This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Symmetries Expand Verified" and has 44 rules.

Found on February 04, 2022.

Finding the specification took 71294 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_{25}\! \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_{25}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \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 , 1\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) F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{26}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{18}\! \left(x \right) &= F_{9} \left(x \right)^{2}\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{25}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{17}\! \left(x , y\right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= x\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{25}\! \left(x \right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{25}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{34}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{25}\! \left(x \right) F_{34}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{37}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{34}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 60 rules.

Found on January 17, 2022.

Finding the specification took 1487 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_{20}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \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_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}\\ F_{34}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{37}\! \left(x \right)+F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{37}\! \left(x \right) &= 0\\ F_{38}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= y x\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{4}\! \left(x \right) F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)^{2} F_{4}\! \left(x \right)\\ F_{49}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{50}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{50}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{52}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{56}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{59}\! \left(x \right) &= F_{54}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 100 rules.

Found on January 17, 2022.

Finding the specification took 718 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_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{16} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \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) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= 0\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x , 1\right)\\ F_{43}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{42}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x , y\right)+F_{98}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{45}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{46}\! \left(x , y\right)+F_{48}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= -\frac{y \left(F_{45}\! \left(x , 1\right)-F_{45}\! \left(x , y\right)\right)}{-1+y}\\ F_{48}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= -\frac{y \left(F_{55}\! \left(x , 1\right)-F_{55}\! \left(x , y\right)\right)}{-1+y}\\ F_{56}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{79}\! \left(x \right)\\ F_{57}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{56}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{77}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{60}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{61}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{62}\! \left(x , y\right)+F_{65}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{6}\! \left(x \right)\\ F_{65}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{42}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{61}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= y x\\ F_{68}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{44}\! \left(x , y\right) F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{1}\! \left(x \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_{67}\! \left(x , y\right) F_{72}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)^{2} F_{4}\! \left(x \right)\\ F_{77}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{44}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{58}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83} \left(x \right)^{2} F_{2}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{83} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{86}\! \left(x , y\right) &= F_{56}\! \left(x , y\right) F_{87}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{87}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{89}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{92}\! \left(x , y\right)+F_{94}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{93}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{8}\! \left(x \right)+F_{93}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= 0\\ F_{95}\! \left(x , y\right) &= F_{67}\! \left(x , y\right) F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{6}\! \left(x \right)+F_{91}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{47}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ \end{align*}\)