Av(12354, 13254, 13524, 13542, 21354, 23154, 23514, 23541, 31254, 31524, 31542, 32154, 32514, 32541, 35124, 35142, 35214, 35241, 35412, 35421)
Generating Function
\(\displaystyle \frac{-6 x^{3} \sqrt{-4 x +1}+2 x^{2} \sqrt{-4 x +1}+16 x^{3}+8 x^{2}-7 x +1}{\left(-1+4 x \right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 420, 1764, 7392, 30888, 128700, 534820, 2217072, 9170616, 37858184, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-1+4 x \right)^{3} F \left(x
\right)^{2}-2 \left(x +1\right) \left(-1+4 x \right)^{3} F \! \left(x \right)+36 x^{6}+40 x^{5}+84 x^{4}-20 x^{3}-25 x^{2}+10 x -1 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{6 \left(2 n - 3\right) a{\left(n \right)}}{n} + \frac{\left(7 n - 4\right) a{\left(n + 1 \right)}}{n}, \quad n \geq 4\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{6 \left(2 n - 3\right) a{\left(n \right)}}{n} + \frac{\left(7 n - 4\right) a{\left(n + 1 \right)}}{n}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 111 rules.
Finding the specification took 670 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_{14}\! \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_{7}\! \left(x \right)\\
F_{6}\! \left(x \right) &= 4 x F_{6} \left(x \right)^{2}+x^{2}-F_{6} \left(x \right)^{2}+F_{6}\! \left(x \right)\\
F_{7}\! \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_{17}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{33}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 4 x F_{41} \left(x \right)^{2}+x^{2}-8 x F_{41}\! \left(x \right)-F_{41} \left(x \right)^{2}+4 x +3 F_{41}\! \left(x \right)-1\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13} \left(x \right)^{3} F_{14}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{13} \left(x \right)^{2}}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= -F_{99}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{14}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{85}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{82}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{2}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{54}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= \frac{F_{88}\! \left(x \right)}{F_{98}\! \left(x \right)}\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{96}\! \left(x \right) &= \frac{F_{97}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{97}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{101}\! \left(x \right) &= -F_{103}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{104}\! \left(x \right) &= -F_{109}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{63}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 128 rules.
Finding the specification took 415 seconds.
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Copy 128 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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{5}\! \left(x \right) &= 4 F_{5} \left(x \right)^{2} x +x^{2}-8 F_{5}\! \left(x \right) x -F_{5} \left(x \right)^{2}+4 x +3 F_{5}\! \left(x \right)-1\\
F_{6}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \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_{17}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{34}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13} \left(x \right)^{3} F_{14}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{13} \left(x \right)^{2}}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= -F_{116}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{14}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{85}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{82}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{2}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{54}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= \frac{F_{88}\! \left(x \right)}{F_{115}\! \left(x \right)}\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= -F_{106}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= -F_{103}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= \frac{F_{93}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{14}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{13}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{64}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= \frac{F_{105}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{105}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{108}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{110}\! \left(x \right) &= \frac{F_{111}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{118}\! \left(x \right) &= -F_{120}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{121}\! \left(x \right) &= -F_{126}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{63}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 8 rules.
Finding the specification took 0 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{6}\! \left(x , y\right) &= y^{2} x^{2}+4 x F_{6}\! \left(x , y\right)^{2} y -8 x F_{6}\! \left(x , y\right) y +4 y x -F_{6}\! \left(x , y\right)^{2}+3 F_{6}\! \left(x , y\right)-1\\
F_{7}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 111 rules.
Finding the specification took 670 seconds.
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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_{14}\! \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_{7}\! \left(x \right)\\
F_{6}\! \left(x \right) &= 4 x F_{6} \left(x \right)^{2}+x^{2}-F_{6} \left(x \right)^{2}+F_{6}\! \left(x \right)\\
F_{7}\! \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_{17}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{33}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 4 F_{41} \left(x \right)^{2} x +x^{2}-8 F_{41}\! \left(x \right) x -F_{41} \left(x \right)^{2}+4 x +3 F_{41}\! \left(x \right)-1\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13} \left(x \right)^{3} F_{14}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{13} \left(x \right)^{2}}\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= -F_{99}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{14}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= -F_{85}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= \frac{F_{79}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{82}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{2}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{54}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= \frac{F_{88}\! \left(x \right)}{F_{98}\! \left(x \right)}\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= -F_{96}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{96}\! \left(x \right) &= \frac{F_{97}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{97}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{101}\! \left(x \right) &= -F_{103}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{104}\! \left(x \right) &= -F_{109}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{63}\! \left(x \right)\\
\end{align*}\)