Av(123, 1432, 2143, 2413, 3142)
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Generating Function
\(\displaystyle -\frac{x -1}{x^{4}-x^{3}-2 x +1}\)
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Counting Sequence
1, 1, 2, 5, 10, 21, 45, 95, 201, 426, 902, 1910, 4045, 8566, 18140, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-x^{3}-2 x +1\right) F \! \left(x \right)+x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)-2 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle -\frac{75 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{643}+\frac{22 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{643}+\frac{127 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{643}+\frac{94 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{643}\)
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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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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\ \end{align*}\)