Av(1243, 2413, 3412)
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Generating Function
\(\displaystyle -\frac{7 x^{5}-27 x^{4}+44 x^{3}-30 x^{2}+9 x -1}{\left(3 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 263, 901, 3024, 9980, 32489, 104585, 333549, 1055497, 3318014, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+7 x^{5}-27 x^{4}+44 x^{3}-30 x^{2}+9 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(n +5\right) = 3 a \! \left(n \right)-19 a \! \left(n +1\right)+39 a \! \left(n +2\right)-29 a \! \left(n +3\right)+9 a \! \left(n +4\right)-2, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{3 \left(3+\sqrt{5}\right) \left(\left(\left(n -\frac{4}{15}\right) \sqrt{5}-\frac{7 n}{3}-\frac{2}{3}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\frac{2 n}{3}+\frac{29 \sqrt{5}}{15}-\frac{13}{3}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-\frac{5 \left(\sqrt{5}-3\right) \left(3^{n}+1\right)}{3}\right)}{20}\)
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This specification was found using the strategy pack "Point And Row Placements" and has 39 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_{12}\! \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_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ 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_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\ \end{align*}\)