Av(1243, 2314, 3142)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right)}{3 x^{4}-14 x^{3}+16 x^{2}-7 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 274, 978, 3463, 12201, 42869, 150415, 527426, 1848905, 6480722, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{4}-14 x^{3}+16 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(3 x -1\right) \left(2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -3 a \! \left(n \right)+14 a \! \left(n +1\right)-16 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \frac{450 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-14 Z^{3}+16 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{491}-\frac{1755 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-14 Z^{3}+16 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{491}+\frac{1300 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-14 Z^{3}+16 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{491}-\frac{217 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-14 Z^{3}+16 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{491}\)
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This specification was found using the strategy pack "Point Placements" and has 32 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{31}\! \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_{28}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \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_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{19}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\ \end{align*}\)