Av(1243, 1342, 2134, 2341, 3124)
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Generating Function
\(\displaystyle \frac{\left(x^{4}-x^{3}-2 x^{2}+2 x -1\right) \sqrt{1-4 x}+2 x^{5}-3 x^{4}-x^{3}+2 x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 174, 549, 1788, 5977, 20395, 70732, 248533, 882716, 3163553, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}-\left(2 x^{5}-3 x^{4}-x^{3}+2 x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}-2 x^{8}-x^{7}+x^{6}+4 x^{5}-x^{4}-5 x^{3}+7 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 549\)
\(\displaystyle a \! \left(8\right) = 1788\)
\(\displaystyle a \! \left(9\right) = 5977\)
\(\displaystyle a \! \left(n +6\right) = \frac{2 \left(-3+2 n \right) a \! \left(n \right)}{7+n}-\frac{\left(-8+9 n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(21+2 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(55+17 n \right) a \! \left(n +3\right)}{7+n}-\frac{8 \left(9+2 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(40+7 n \right) a \! \left(n +5\right)}{7+n}+\frac{2}{7+n}, \quad n \geq 10\)
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This specification was found using the strategy pack "Pattern Point Placements Expand Verified" and has 35 rules.

Found on January 21, 2022.

Finding the specification took 194 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{13}\! \left(x , y\right) &= \frac{y F_{14}\! \left(x , y\right)-F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{25}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30} \left(x \right)^{2}\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Expand Verified" and has 75 rules.

Found on January 21, 2022.

Finding the specification took 32 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{13}\! \left(x , y\right) &= -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}\\ F_{14}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= y x\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 0\\ F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{10}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{22}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= -F_{32}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{62}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= -F_{30}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{10}\! \left(x \right)}\\ F_{72}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{62}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements Req Corrob Expand Verified" and has 94 rules.

Found on January 21, 2022.

Finding the specification took 115 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_{1}\! \left(x \right)+F_{5}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{5}\! \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) &= \frac{F_{10}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{17}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{89}\! \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_{70}\! \left(x \right)-F_{71}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{23}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \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_{1}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= -F_{52}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{51}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{57}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{24}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{23}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 0\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= \frac{F_{73}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= -F_{79}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{14}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{14}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= \frac{F_{90}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{90}\! \left(x \right) &= -F_{91}\! \left(x \right)-2 F_{70}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{14}\! \left(x \right) F_{41}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{14}\! \left(x \right) F_{75}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Row And Col Placements Req Corrob Expand Verified" and has 45 rules.

Found on January 21, 2022.

Finding the specification took 82 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \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) &= \frac{F_{9}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{9}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{10}\! \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_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= -\frac{-y F_{18}\! \left(x , y\right)+F_{18}\! \left(x , 1\right)}{-1+y}\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= y x\\ F_{21}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{17}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{26}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{28}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{39}\! \left(x \right)-F_{41}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{8}\! \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_{8}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \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_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements Expand Verified" and has 67 rules.

Found on January 21, 2022.

Finding the specification took 23 seconds.

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Copy 67 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_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \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_{8}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{14}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{14}\! \left(x , y\right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{11}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{23}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= -F_{33}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= \frac{F_{6}\! \left(x \right)}{F_{11}\! \left(x \right)}\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{31} \left(x \right)^{2}\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{33}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{56}\! \left(x \right) &= -F_{63}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= -F_{62}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{11}\! \left(x \right)}\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= -F_{7}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{31}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{57}\! \left(x \right)\\ \end{align*}\)