Av(1342, 2143, 2413)
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Generating Function
\(\displaystyle \frac{x^{2}-3 x +1-\sqrt{-8 x^{5}+37 x^{4}-54 x^{3}+35 x^{2}-10 x +1}}{4 x^{3}-6 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 288, 1093, 4203, 16359, 64377, 255857, 1025889, 4145966, 16873475, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right) \left(x -1\right) F \left(x \right)^{2}+\left(-x^{2}+3 x -1\right) F \! \left(x \right)+x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 288\)
\(\displaystyle a \! \left(n +7\right) = \frac{8 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(80+49 n \right) a \! \left(1+n \right)}{n +8}+\frac{\left(604+227 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(991+269 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(844+179 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(386+67 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(89+13 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements" and has 19 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ \end{align*}\)