Av(1342, 1423, 2314, 2341, 3412)
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Generating Function
\(\displaystyle -\frac{3 x^{6}-18 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 161, 435, 1143, 2958, 7604, 19517, 50154, 129195, 333712, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{6}-18 x^{5}+38 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 161\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-\frac{\left(n -2\right) \left(n^{2}+2 n +3\right)}{6}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{n^{3}}{6}-\frac{11 n}{6}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+2^{n}\)
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This specification was found using the strategy pack "Point Placements" and has 45 rules.

Found on July 23, 2021.

Finding the specification took 11 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_{7}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{29}\! \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_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{11}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{33}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{42}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right)\\ \end{align*}\)