Av(1243, 1342, 2413)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1+\sqrt{4 x^{4}-20 x^{3}+20 x^{2}-8 x +1}\right) \left(2 x -1\right)}{2 x \left(x^{3}-5 x^{2}+4 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 287, 1082, 4128, 15945, 62330, 246328, 982977, 3956136, 16041373, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{3}-5 x^{2}+4 x -1\right) F \left(x \right)^{2}+\left(2 x -1\right)^{2} F \! \left(x \right)-\left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 287\)
\(\displaystyle a \! \left(7\right) = 1082\)
\(\displaystyle a \! \left(n +8\right) = -\frac{8 \left(1+n \right) a \! \left(n \right)}{9+n}+\frac{4 \left(21 n +38\right) a \! \left(1+n \right)}{9+n}-\frac{2 \left(156 n +421\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(260 n +957\right) a \! \left(n +3\right)}{9+n}-\frac{6 \left(79 n +371\right) a \! \left(n +4\right)}{9+n}+\frac{3 \left(85 n +487\right) a \! \left(n +5\right)}{9+n}-\frac{9 \left(9 n +61\right) a \! \left(n +6\right)}{9+n}+\frac{2 \left(7 n +55\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point Placements" and has 33 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\ F_{15}\! \left(x \right) &= F_{11}\! \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_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 0\\ F_{30}\! \left(x \right) &= F_{19}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\ \end{align*}\)