Av(1342, 1423, 1432, 3241)
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Generating Function
\(\displaystyle \frac{x^{4}+2 x^{3}+x^{2}+2 x -1}{x^{4}+2 x^{3}+3 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 209, 673, 2169, 6990, 22525, 72586, 233907, 753761, 2428980, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+2 x^{3}+3 x -1\right) F \! \left(x \right)-x^{4}-2 x^{3}-x^{2}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +3\right)}{2}+\frac{a \! \left(n +4\right)}{2}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{46 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+2 Z^{3}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{935}+\frac{35 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+2 Z^{3}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{187}+\frac{19 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+2 Z^{3}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{935}+\frac{6 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+2 Z^{3}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{187}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 79 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{30}\! \left(x \right) &= 0\\ F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{10}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{10}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{10}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{60}\! \left(x \right) &= x^{2}\\ F_{61}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{10}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{46}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{10}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{10}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 3 F_{30}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{10}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{10}\! \left(x \right) F_{45}\! \left(x \right)\\ \end{align*}\)