Av(123, 1432, 2431, 3241, 4213)
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Generating Function
\(\displaystyle \frac{2 x^{7}-x^{6}-6 x^{5}+x^{4}+2 x^{3}-x +1}{\left(x -1\right) \left(x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 5, 10, 12, 18, 28, 44, 70, 112, 180, 290, 468, 756, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+2 x^{7}-x^{6}-6 x^{5}+x^{4}+2 x^{3}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 10\)
\(\displaystyle a \! \left(5\right) = 12\)
\(\displaystyle a \! \left(6\right) = 18\)
\(\displaystyle a \! \left(7\right) = 28\)
\(\displaystyle a \! \left(n +2\right) = a \! \left(n \right)+a \! \left(n +1\right)-2, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\left(\left\{\begin{array}{cc}1 & n =0 \\ 3 & n =1 \\ 2 & n =2 \\ 1 & n =3 \\ -2 & n =4 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{8 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}+\frac{8 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{1-n}}{5}+\frac{\left(2 \sqrt{5}+4\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+2+\frac{\left(-2 \sqrt{5}+4\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}\)
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This specification was found using the strategy pack "Point Placements" and has 36 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \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_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \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_{6}\! \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_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{4}\! \left(x \right)\\ \end{align*}\)