Av(123, 2143, 2413, 3241, 4132)
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Generating Function
\(\displaystyle \frac{x^{8}-x^{7}-3 x^{6}+3 x^{5}+x^{4}-2 x^{3}-x^{2}+2 x -1}{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 5, 10, 16, 28, 48, 81, 135, 223, 366, 598, 974, 1583, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-x^{7}-3 x^{6}+3 x^{5}+x^{4}-2 x^{3}-x^{2}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 10\)
\(\displaystyle a \! \left(5\right) = 16\)
\(\displaystyle a \! \left(6\right) = 28\)
\(\displaystyle a \! \left(7\right) = 48\)
\(\displaystyle a \! \left(8\right) = 81\)
\(\displaystyle a \! \left(n +2\right) = a \! \left(n \right)+a \! \left(n +1\right)-1+n, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \left(\left\{\begin{array}{cc}-1 & n =0\text{ or } n =1\text{ or } n =2 \\ 1 & n =4 \\ 0 & \text{otherwise} \end{array}\right.\right)-n -\frac{2 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\frac{2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}+\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}\)
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This specification was found using the strategy pack "Point Placements" and has 32 rules.

Found on January 18, 2022.

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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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ 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_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= x^{2}\\ F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{29}\! \left(x \right)\\ \end{align*}\)