Av(1324, 1342, 3412)
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Generating Function
\(\displaystyle -\frac{11 x^{6}-50 x^{5}+88 x^{4}-81 x^{3}+40 x^{2}-10 x +1}{\left(3 x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 261, 879, 2879, 9232, 29148, 90995, 281730, 866917, 2655218, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+11 x^{6}-50 x^{5}+88 x^{4}-81 x^{3}+40 x^{2}-10 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 261\)
\(\displaystyle a \! \left(n +4\right) = -6 a \! \left(n \right)+23 a \! \left(n +1\right)-22 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{n \left(n +1\right)}{2}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{n^{2}}{4}+\frac{n}{2}+\frac{11}{8}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-2^{n}+\frac{5 \,3^{n}}{8}\)
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This specification was found using the strategy pack "Row And Col Placements" and has 50 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_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \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_{7}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \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) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 2 F_{39}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{29}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{10}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right) F_{44}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{10}\! \left(x \right) F_{45}\! \left(x \right)\\ \end{align*}\)