Av(1243, 1324, 2143, 2413, 2431)
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Generating Function
\(\displaystyle \frac{\left(-x^{2}-x +1\right) \sqrt{1-4 x}+3 x^{2}+x -1}{4 x^{2}-2 x}\)
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Counting Sequence
1, 1, 2, 6, 19, 61, 198, 651, 2171, 7345, 25194, 87516, 307462, 1091026, 3905580, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} x F \left(x \right)^{2}-\left(2 x -1\right) \left(3 x^{2}+x -1\right) F \! \left(x \right)+x^{4}+4 x^{3}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +5}+\frac{2 \left(6+n \right) a \! \left(1+n \right)}{n +5}-\frac{13 \left(n +3\right) a \! \left(n +2\right)}{n +5}+\frac{\left(27+7 n \right) a \! \left(n +3\right)}{n +5}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements" and has 21 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ \end{align*}\)