Av(1243, 2314, 2341, 4213)
Generating Function
\(\displaystyle \frac{5 x^{9}-4 x^{8}-11 x^{7}+12 x^{6}-16 x^{4}+27 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 61, 162, 386, 856, 1814, 3740, 7590, 15273, 30606, 61226, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+5 x^{9}-4 x^{8}-11 x^{7}+12 x^{6}-16 x^{4}+27 x^{3}-20 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(7\right) = 386\)
\(\displaystyle a \! \left(8\right) = 856\)
\(\displaystyle a \! \left(9\right) = 1814\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{85 n^{2}}{24}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{275 n}{12}-18, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 162\)
\(\displaystyle a \! \left(7\right) = 386\)
\(\displaystyle a \! \left(8\right) = 856\)
\(\displaystyle a \! \left(9\right) = 1814\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{85 n^{2}}{24}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{275 n}{12}-18, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(36 \sqrt{5}-60\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{\left(-36 \sqrt{5}-60\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{12}\\-\frac{85 n^{2}}{24}+\frac{91 n}{12}+\frac{15 \,2^{n}}{4}-11 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 79 rules.
Found on January 18, 2022.Finding the specification took 1 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{29}\! \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_{21}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{40}\! \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_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{68}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{67}\! \left(x \right)\\
\end{align*}\)