Av(1243, 1324, 2413, 3241)
Generating Function
\(\displaystyle \frac{x^{8}-7 x^{7}+31 x^{6}-74 x^{5}+106 x^{4}-88 x^{3}+41 x^{2}-10 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 194, 565, 1603, 4473, 12349, 33853, 92351, 251044, 680606, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)-x^{8}+7 x^{7}-31 x^{6}+74 x^{5}-106 x^{4}+88 x^{3}-41 x^{2}+10 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 194\)
\(\displaystyle a \! \left(7\right) = 565\)
\(\displaystyle a \! \left(8\right) = 1603\)
\(\displaystyle a \! \left(n +5\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-\frac{n}{2}-2, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 194\)
\(\displaystyle a \! \left(7\right) = 565\)
\(\displaystyle a \! \left(8\right) = 1603\)
\(\displaystyle a \! \left(n +5\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-\frac{n}{2}-2, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-15 n -21\right) \sqrt{5}+35 n +75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(15 n +21\right) \sqrt{5}+35 n +75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-\frac{n^{2}}{2}+\frac{3 n}{2}-2^{-1+n}-2 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 77 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \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_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{31}\! \left(x \right) &= 0\\
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_{41}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \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_{10}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \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_{55}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{61}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
\end{align*}\)