Av(1243, 1342, 3142)
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Generating Function
\(\displaystyle \frac{-x^{2}+2 x -1+\sqrt{x^{4}-12 x^{3}+18 x^{2}-8 x +1}}{2 x \left(x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 78, 299, 1172, 4677, 18947, 77746, 322545, 1350906, 5704822, 24265651, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right) F \left(x \right)^{2}+\left(x -1\right)^{2} F \! \left(x \right)+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n a \! \left(n \right)}{5+n}+\frac{\left(17+12 n \right) a \! \left(1+n \right)}{5+n}-\frac{\left(43+18 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(29+8 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements" and has 14 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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