Av(123, 1432, 2413, 2431, 3214)
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Generating Function
\(\displaystyle -\frac{2 x^{4}+x^{3}+1}{x^{3}+x^{2}+x -1}\)
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Counting Sequence
1, 1, 2, 5, 10, 17, 32, 59, 108, 199, 366, 673, 1238, 2277, 4188, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{4}+x^{3}+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 10\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{5 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\left(\left(i+\frac{29 \sqrt{11}}{165}\right) \sqrt{3}-\frac{29 i \sqrt{11}}{55}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\frac{i}{5}-\frac{\sqrt{11}}{165}\right) \sqrt{3}-\frac{i \sqrt{11}}{55}-\frac{1}{5}\right) \left(\frac{\left(\left(17 i+3 \sqrt{11}\right) \sqrt{3}-9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(\left(\left(-i+\frac{29 \sqrt{11}}{165}\right) \sqrt{3}+\frac{29 i \sqrt{11}}{55}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\frac{i}{5}-\frac{\sqrt{11}}{165}\right) \sqrt{3}+\frac{i \sqrt{11}}{55}-\frac{1}{5}\right) \left(\frac{\left(\left(-17 i+3 \sqrt{11}\right) \sqrt{3}+9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}-\frac{58 \left(\left(\sqrt{11}\, \sqrt{3}-\frac{165}{29}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{\sqrt{11}\, \sqrt{3}}{29}-\frac{33}{29}\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{165}\right)}{8} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 22 rules.

Found on January 18, 2022.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)