Av(1243, 2314, 2341, 3142, 3214)
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Generating Function
\(\displaystyle \frac{x^{7}+x^{6}+3 x^{5}-3 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 142, 359, 887, 2158, 5193, 12396, 29412, 69468, 163504, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)-x^{7}-x^{6}-3 x^{5}+3 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 142\)
\(\displaystyle a \! \left(7\right) = 359\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-1, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{12461 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{2783}-\frac{19438 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{2783}+\frac{56677 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{5566}-\frac{23910 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{2783}+\frac{9357 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{506}-\frac{152483 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{5566}+\frac{25935 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-2 Z^{6}+3 Z^{5}-3 Z^{4}+5 Z^{3}-8 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{2783}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 40 rules.

Found on January 18, 2022.

Finding the specification took 0 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_{24}\! \left(x \right)+F_{6}\! \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_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \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_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \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_{18}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{25}\! \left(x \right)\\ \end{align*}\)