Av(13452, 14352, 14532, 31452, 34152, 34512, 41352, 41532, 43152, 43512, 45132, 45312)
Generating Function
\(\displaystyle \frac{2 x^{3}-5 x^{2}-\sqrt{12 x^{2}-8 x +1}+5 x +1}{x \left(2 x -3\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2506, 12560, 64148, 332704, 1747748, 9280416, 49731768, 268613568, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -3\right)^{2} F \left(x
\right)^{2}+\left(-4 x^{3}+10 x^{2}-10 x -2\right) F \! \left(x \right)+x^{3}-2 x^{2}+3 x +2 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(n + 2\right) a{\left(n \right)}}{n + 4} - \frac{4 \left(13 n + 23\right) a{\left(n + 1 \right)}}{3 \left(n + 4\right)} + \frac{2 \left(13 n + 35\right) a{\left(n + 2 \right)}}{3 \left(n + 4\right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(n + 2\right) a{\left(n \right)}}{n + 4} - \frac{4 \left(13 n + 23\right) a{\left(n + 1 \right)}}{3 \left(n + 4\right)} + \frac{2 \left(13 n + 35\right) a{\left(n + 2 \right)}}{3 \left(n + 4\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 111 rules.
Finding the specification took 2312 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 111 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_{20}\! \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_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{110}\! \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_{11}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
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_{19}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{20}\! \left(x \right)\\
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_{20}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= y x\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{19}\! \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_{22}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= -\frac{-y F_{31}\! \left(x , y\right)+F_{31}\! \left(x , 1\right)}{-1+y}\\
F_{33}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{32}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{5}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= -\frac{y \left(F_{38}\! \left(x , 1\right)-F_{38}\! \left(x , y\right)\right)}{-1+y}\\
F_{38}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{20}\! \left(x \right) F_{43}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{48}\! \left(x \right) &= 0\\
F_{49}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{46}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= y F_{53}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{55}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{55}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{20}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{20}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{20}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{20}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{78}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{78}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right) F_{85}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= y F_{83}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{20}\! \left(x \right) F_{60}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{64}\! \left(x \right)+F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{89}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{87}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{67}\! \left(x \right)+F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{95}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{93}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{70}\! \left(x \right)+F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{102}\! \left(x , y\right)+F_{104}\! \left(x , y\right)+F_{48}\! \left(x \right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{20}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{20}\! \left(x \right)\\
F_{103}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{98}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= y F_{106}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{20}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{109}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{110}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 52 rules.
Finding the specification took 1736 seconds.
Copy 52 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_{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_{50}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \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_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{38}\! \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_{41}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{33}\! \left(x \right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{47}\! \left(x \right) &= x^{2} F_{47} \left(x \right)^{2}+2 x^{2} F_{47}\! \left(x \right)-2 x F_{47} \left(x \right)^{2}+x^{2}-3 x F_{47}\! \left(x \right)-x +2 F_{47}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 49 rules.
Finding the specification took 903 seconds.
Copy 49 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_{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_{47}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \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_{24}\! \left(x \right)\\
F_{21}\! \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_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{36}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{31}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{11}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 49 rules.
Finding the specification took 903 seconds.
Copy 49 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_{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_{47}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{29}\! \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_{24}\! \left(x \right)\\
F_{21}\! \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_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{36}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{31}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{11}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 52 rules.
Finding the specification took 1736 seconds.
Copy 52 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_{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_{50}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}-2 x F_{11} \left(x \right)^{2}+F_{11}\! \left(x \right) x +2 F_{11}\! \left(x \right)-1\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \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_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{38}\! \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_{41}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{33}\! \left(x \right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{47}\! \left(x \right) &= x^{2} F_{47} \left(x \right)^{2}+2 x^{2} F_{47}\! \left(x \right)-2 x F_{47} \left(x \right)^{2}+x^{2}-3 x F_{47}\! \left(x \right)-x +2 F_{47}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)