Av(1243, 1342, 2143, 3124, 3214)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)}{\left(x^{2}-x +1\right) \left(x^{3}-3 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 161, 462, 1329, 3827, 11021, 31735, 91377, 263108, 757588, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) \left(x^{3}-3 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{2 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{27}-\frac{2 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{27}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{9}+\frac{4 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{27}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{27}\)
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This specification was found using the strategy pack "Point Placements" and has 42 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_{17}\! \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_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{33}\! \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_{11}\! \left(x \right)+F_{23}\! \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_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{27}\! \left(x \right)\\ \end{align*}\)