Av(12354, 13254, 13524, 31254, 31524, 35124)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3297, 18921, 111457, 669673, 4087147, 25265542, 157858622, 995263826, ...
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 32 rules.
Finding the specification took 2523 seconds.
Copy 32 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_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= x^{2} F_{6}\! \left(x , y\right)^{2} y^{2}+y x F_{6}\! \left(x , y\right)^{4}-x F_{6}\! \left(x , y\right)^{3} y -2 x F_{6}\! \left(x , y\right)^{2} y +2 x F_{6}\! \left(x , y\right) y -F_{6}\! \left(x , y\right)^{3}+3 F_{6}\! \left(x , y\right)^{2}-2 F_{6}\! \left(x , y\right)+1\\
F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{31}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , 1, y\right)\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y z , z\right)\\
F_{16}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{31}\! \left(x \right)\\
F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)+F_{5}\! \left(x , y\right)\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , z\right)\\
F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{26}\! \left(x , z\right) F_{27}\! \left(x , z\right)\\
F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right)+F_{30}\! \left(x , y , z\right)\\
F_{22}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right) F_{5}\! \left(x , y\right)\\
F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)+F_{6}\! \left(x , y\right)\\
F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= F_{23}\! \left(x , y , z\right) F_{26}\! \left(x , z\right) F_{27}\! \left(x , z\right) F_{6}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= y x\\
F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)^{2} F_{26}\! \left(x , y\right)\\
F_{30}\! \left(x , y , z\right) &= F_{0}\! \left(x \right) F_{17}\! \left(x , y , z\right)\\
F_{31}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 35 rules.
Finding the specification took 1343 seconds.
Copy 35 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_{34}\! \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 , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+x F_{7}\! \left(x , y\right)^{4} y +2 x^{2} F_{7}\! \left(x , y\right) y^{2}+3 x F_{7}\! \left(x , y\right)^{3} y +x^{2} y^{2}+x F_{7}\! \left(x , y\right)^{2} y -x F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , y\right)^{3}+F_{7}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{34}\! \left(x \right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\
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_{26}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{34}\! \left(x \right)\\
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_{18}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{8}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{26}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= y x\\
F_{28}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{16}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)^{2} F_{26}\! \left(x , y\right)\\
F_{34}\! \left(x \right) &= x\\
\end{align*}\)