Av(1243, 1342, 1423, 1432, 2143, 2413, 2431, 3142, 4132, 4231)
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Generating Function
\(\displaystyle -\frac{4 x^{5}-4 x^{4}+5 x^{2}-4 x +1}{\left(x -1\right) \left(-1+2 x \right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 14, 34, 82, 194, 450, 1026, 2306, 5122, 11266, 24578, 53250, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(-1+2 x \right)^{2} F \! \left(x \right)+4 x^{5}-4 x^{4}+5 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 14\)
\(\displaystyle a \! \left(5\right) = 34\)
\(\displaystyle a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+2, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 2+\frac{\left(-1+n \right) 2^{n}}{4} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 18 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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