Av(1324, 2341, 2431, 3241, 3412)
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Generating Function
\(\displaystyle -\frac{x^{7}+x^{5}+x^{4}+5 x^{2}-4 x +1}{\left(x -1\right) \left(2 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 152, 389, 953, 2261, 5237, 11909, 26693, 59141, 129797, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{7}+x^{5}+x^{4}+5 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 152\)
\(\displaystyle a \! \left(7\right) = 389\)
\(\displaystyle a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+5, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ 5+\frac{\left(45 n -123\right) 2^{n}}{64} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 36 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_{16}\! \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_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \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) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\ \end{align*}\)