Av(13524, 13542, 31524, 31542)
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Generating Function
\(\displaystyle \frac{-x \sqrt{-8 x +1}-x +2}{4 x^{2}-4 x +2}\)
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Counting Sequence
1, 1, 2, 6, 24, 116, 632, 3720, 23072, 148528, 983072, 6647776, 45727616, 318947136, 2250473344, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-2 x +1\right) F \left(x \right)^{2}+\left(x -2\right) F \! \left(x \right)+x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{6 \left(3 n +2\right) a \! \left(n +1\right)}{n +2}+\frac{2 \left(5 n +4\right) a \! \left(n +2\right)}{n +2}, \quad n \geq 3\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 25 rules.

Found on January 22, 2022.

Finding the specification took 12 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)