Av(1342, 3412)
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Generating Function
\(\displaystyle \frac{\left(2 \sqrt{5 x^{2}-6 x +1}\, x -4 x^{2}-\sqrt{5 x^{2}-6 x +1}+3 x \right) \left(x -1\right)}{4 x^{4}-20 x^{3}+24 x^{2}-10 x +1}\)
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Counting Sequence
1, 1, 2, 6, 22, 88, 366, 1556, 6720, 29396, 129996, 580276, 2611290, 11834116, 53963190, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{4}-20 x^{3}+24 x^{2}-10 x +1\right) F \left(x \right)^{2}+2 x \left(4 x -3\right) \left(x -1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 88\)
\(\displaystyle a \! \left(6\right) = 366\)
\(\displaystyle a \! \left(n +7\right) = \frac{40 \left(n +1\right) a \! \left(n \right)}{n +7}-\frac{4 \left(105+67 n \right) a \! \left(n +1\right)}{n +7}+\frac{4 \left(377+153 n \right) a \! \left(n +2\right)}{n +7}-\frac{16 \left(143+42 n \right) a \! \left(n +3\right)}{n +7}+\frac{28 \left(61+14 n \right) a \! \left(n +4\right)}{n +7}-\frac{\left(643+121 n \right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(56+9 n \right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7\)
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Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).

This specification was found using the strategy pack "Point And Col Placements" and has 26 rules.

Found on January 17, 2022.

Finding the specification took 98 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ 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_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \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_{3}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \left(x \right) F_{3}\! \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_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Row And Col Placements" and has 54 rules.

Found on January 18, 2022.

Finding the specification took 34 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_{8}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \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_{6}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{0}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \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_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= 0\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{34}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{35}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{19}\! \left(x \right) F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{49} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{39}\! \left(x \right) F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements" and has 59 rules.

Found on January 18, 2022.

Finding the specification took 39 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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{22}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \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_{10}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ 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_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{26}\! \left(x \right) &= -F_{50}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= -F_{4}\! \left(x \right)-F_{43}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{40}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{40} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{45} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{45}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{45}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{7}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements Req Corrob" and has 77 rules.

Found on January 17, 2022.

Finding the specification took 174 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_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \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_{22}\! \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) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{2}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 0\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{2}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{24}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{18}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right) F_{6}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{24}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{57}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Req Corrob" and has 90 rules.

Found on January 17, 2022.

Finding the specification took 147 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_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \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_{22}\! \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) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{18}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{46}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 0\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{24}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{18}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{24}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)