Av(1243, 2143, 2431, 3412)
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Generating Function
\(\displaystyle -\frac{x^{8}-7 x^{7}+15 x^{6}-26 x^{5}+50 x^{4}-56 x^{3}+32 x^{2}-9 x +1}{\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 62, 182, 518, 1449, 4012, 11036, 30220, 82474, 224490, 609743, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{8}-7 x^{7}+15 x^{6}-26 x^{5}+50 x^{4}-56 x^{3}+32 x^{2}-9 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 182\)
\(\displaystyle a \! \left(7\right) = 518\)
\(\displaystyle a \! \left(8\right) = 1449\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)+n, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\left(-30 n -2\right) \sqrt{5}+70 n +50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{100}+\\\frac{\left(\left(30 n +2\right) \sqrt{5}+70 n +50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{100}-n -\frac{2^{n}}{4}+1 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Row Placements" and has 50 rules.

Found on July 23, 2021.

Finding the specification took 5 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{6}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \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_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{14}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 0\\ F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\ \end{align*}\)