Av(1243, 1342, 1423, 1432, 2143, 2413)
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Generating Function
\(\displaystyle \frac{1-\sqrt{-4 x^{3}-4 x +1}}{2 x \left(x^{2}+1\right)}\)
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Counting Sequence
1, 1, 2, 6, 18, 57, 190, 654, 2306, 8290, 30272, 111973, 418666, 1579803, 6008464, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+1\right) F \left(x \right)^{2}-F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(n \right) = -\frac{4 \left(n +3\right) a \! \left(n +2\right)}{3+2 n}+\frac{\left(6+n \right) a \! \left(n +3\right)}{6+4 n}-\frac{\left(9+2 n \right) a \! \left(n +4\right)}{3+2 n}+\frac{\left(6+n \right) a \! \left(n +5\right)}{6+4 n}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements" and has 10 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ \end{align*}\)