Av(1324, 1342, 1432, 4123, 4213)
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Generating Function
\(\displaystyle \frac{3 x^{5}+6 x^{4}+2 x^{3}-x^{2}-2 x +1}{\left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 52, 137, 353, 892, 2227, 5512, 13556, 33185, 80958, 196999, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)-3 x^{5}-6 x^{4}-2 x^{3}+x^{2}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{135 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{176}-\frac{11 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{4}-\frac{32 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{11}-\frac{167 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{88}+\frac{329 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}-3 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{176}+\left(\left\{\begin{array}{cc}3 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 61 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{13}\! \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_{6}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{26}\! \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_{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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \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_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{27}\! \left(x \right) &= 0\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= x^{2}\\ F_{39}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{57}\! \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_{56}\! \left(x \right) &= x^{2}\\ F_{57}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)