Av(1243, 1342, 1423, 2134, 2314, 4123)
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Generating Function
\(\displaystyle \frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right)}{2 x \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 18, 51, 152, 477, 1557, 5228, 17920, 62395, 219969, 783433, 2814269, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x \right)^{2}+\left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 152\)
\(\displaystyle a \! \left(7\right) = 477\)
\(\displaystyle a \! \left(8\right) = 1557\)
\(\displaystyle a \! \left(9\right) = 5228\)
\(\displaystyle a \! \left(10\right) = 17920\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(-5+2 n \right) a \! \left(n \right)}{8+n}-\frac{\left(-17+9 n \right) a \! \left(1+n \right)}{8+n}+\frac{\left(-13+2 n \right) a \! \left(n +2\right)}{8+n}-\frac{9 a \! \left(n +3\right)}{8+n}+\frac{3 \left(19+4 n \right) a \! \left(n +4\right)}{8+n}-\frac{3 \left(28+5 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(47+7 n \right) a \! \left(n +6\right)}{8+n}+\frac{1}{8+n}, \quad n \geq 11\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 31 rules.

Found on July 23, 2021.

Finding the specification took 9 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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \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_{17}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \left(x \right)\\ \end{align*}\)