Av(1243, 1432, 2314)
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Generating Function
\(\displaystyle \frac{\left(3 x -1\right) \left(-1+x \right)^{3}}{9 x^{4}-19 x^{3}+17 x^{2}-7 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 74, 257, 886, 3050, 10505, 36206, 124833, 430474, 1484526, 5119597, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(9 x^{4}-19 x^{3}+17 x^{2}-7 x +1\right) F \! \left(x \right)-\left(3 x -1\right) \left(-1+x \right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +4\right) = -9 a \! \left(n \right)+19 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{288 \left(\underset{\alpha =\mathit{RootOf} \left(9 Z^{4}-19 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{491}+\frac{341 \left(\underset{\alpha =\mathit{RootOf} \left(9 Z^{4}-19 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{491}-\frac{100 \left(\underset{\alpha =\mathit{RootOf} \left(9 Z^{4}-19 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{491}+\frac{29 \left(\underset{\alpha =\mathit{RootOf} \left(9 Z^{4}-19 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{491}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 65 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\ \end{align*}\)