Av(1243, 1324, 2413, 2431, 4132)
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Generating Function
\(\displaystyle \frac{\left(6 x^{3}-9 x^{2}+5 x -1\right) \sqrt{1-4 x}+2 x^{5}-6 x^{3}+9 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 175, 537, 1699, 5552, 18660, 64135, 224275, 794814, 2846552, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{5}-6 x^{3}+9 x^{2}-5 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}-6 x^{7}+45 x^{6}-113 x^{5}+142 x^{4}-102 x^{3}+43 x^{2}-10 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 175\)
\(\displaystyle a \! \left(7\right) = 537\)
\(\displaystyle a \! \left(8\right) = 1699\)
\(\displaystyle a \! \left(n +6\right) = -\frac{24 \left(2 n +3\right) a \! \left(n \right)}{7+n}+\frac{12 \left(13 n +25\right) a \! \left(1+n \right)}{7+n}-\frac{2 \left(104 n +281\right) a \! \left(n +2\right)}{7+n}+\frac{49 \left(3 n +11\right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(29 n +137\right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(6 n +35\right) a \! \left(n +5\right)}{7+n}-\frac{n +3}{7+n}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point Placements" and has 29 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_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ \end{align*}\)