Av(1243, 1342, 2143, 2413, 3142)
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Generating Function
\(\displaystyle \frac{\left(-x +1\right) \sqrt{x^{4}+2 x^{3}+7 x^{2}-6 x +1}+x^{3}+2 x -1}{4 x^{2}-2 x}\)
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Counting Sequence
1, 1, 2, 6, 19, 62, 208, 715, 2510, 8971, 32557, 119700, 445003, 1670193, 6320360, ...
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Implicit Equation for the Generating Function
\(\displaystyle -x \left(2 x -1\right) F \left(x \right)^{2}+\left(x^{3}+2 x -1\right) F \! \left(x \right)-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(n -1\right) a \! \left(n \right)}{7+n}-\frac{\left(1+n \right) a \! \left(1+n \right)}{7+n}-\frac{9 \left(n +2\right) a \! \left(n +2\right)}{7+n}+\frac{\left(107+31 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(125+27 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{7+n}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 16 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)