Av(1243, 1342, 1423, 1432, 2143, 2413, 3142)
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Generating Function
\(\displaystyle \frac{-x^{3}+1-\sqrt{x^{6}-2 x^{3}-4 x +1}}{2 x}\)
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Counting Sequence
1, 1, 2, 6, 17, 52, 168, 561, 1922, 6719, 23871, 85938, 312823, 1149421, 4257460, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{2}+\left(x -1\right) \left(x^{2}+x +1\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(n \right) = \frac{\left(2 n +5\right) a \! \left(n +3\right)}{n -2}+\frac{2 \left(2 n +11\right) a \! \left(n +5\right)}{n -2}-\frac{\left(7+n \right) a \! \left(n +6\right)}{n -2}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 9 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{5}\! \left(x \right)\\ \end{align*}\)