Av(1234, 1243, 1342, 2134, 2314, 2341)
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Generating Function
\(\displaystyle \frac{-\sqrt{-4 x +1}\, x^{2}+3 x^{2}+\sqrt{-4 x +1}-1}{2 \left(x -1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 18, 55, 173, 560, 1858, 6291, 21657, 75581, 266797, 950911, 3417339, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{2} F \left(x \right)^{2}-\left(x -1\right) \left(3 x^{2}-1\right) F \! \left(x \right)+x^{4}+2 x^{3}-2 x^{2}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(n +2\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +3}+\frac{\left(5+3 n \right) a \! \left(n +1\right)}{n +3}+\frac{6 n}{n +3}, \quad n \geq 5\)
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This specification was found using the strategy pack "Row And Col Placements Req Corrob Expand Verified" and has 23 rules.

Found on January 21, 2022.

Finding the specification took 22 seconds.

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

This specification was found using the strategy pack "All The Strategies 2 Expand Verified" and has 26 rules.

Found on January 21, 2022.

Finding the specification took 27 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Expand Verified" and has 33 rules.

Found on January 21, 2022.

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

This specification was found using the strategy pack "Point Placements Expand Verified" and has 27 rules.

Found on January 21, 2022.

Finding the specification took 37 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_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{17}\! \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_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= -\frac{-y F_{13}\! \left(x , y\right)+F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= \frac{F_{23}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{23}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \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) F_{9}\! \left(x \right)\\ \end{align*}\)