Av(1243, 3214, 3241, 4213)
Generating Function
\(\displaystyle -\frac{x^{12}-12 x^{11}+44 x^{10}-45 x^{9}-87 x^{8}+346 x^{7}-598 x^{6}+665 x^{5}-494 x^{4}+241 x^{3}-74 x^{2}+13 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 62, 174, 457, 1158, 2887, 7159, 17751, 44110, 109942, 274939, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{5} F \! \left(x \right)+x^{12}-12 x^{11}+44 x^{10}-45 x^{9}-87 x^{8}+346 x^{7}-598 x^{6}+665 x^{5}-494 x^{4}+241 x^{3}-74 x^{2}+13 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 457\)
\(\displaystyle a \! \left(8\right) = 1158\)
\(\displaystyle a \! \left(9\right) = 2887\)
\(\displaystyle a \! \left(10\right) = 7159\)
\(\displaystyle a \! \left(11\right) = 17751\)
\(\displaystyle a \! \left(12\right) = 44110\)
\(\displaystyle a \! \left(n +5\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{37 n^{2}}{24}+8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)+\frac{143 n}{12}-3, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 457\)
\(\displaystyle a \! \left(8\right) = 1158\)
\(\displaystyle a \! \left(9\right) = 2887\)
\(\displaystyle a \! \left(10\right) = 7159\)
\(\displaystyle a \! \left(11\right) = 17751\)
\(\displaystyle a \! \left(12\right) = 44110\)
\(\displaystyle a \! \left(n +5\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{37 n^{2}}{24}+8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)+\frac{143 n}{12}-3, \quad n \geq 13\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-96 \sqrt{5}+480\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{960}+\frac{\left(96 \sqrt{5}+480\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{960}+\\\frac{\left(15 n^{2}+135 n -300\right) 2^{n}}{960}+\frac{n^{4}}{24}-\frac{n^{3}}{12}-\frac{25 n^{2}}{24}+\frac{37 n}{12}-2 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 89 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_{28}\! \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_{20}\! \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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 0\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{84}\! \left(x \right)+F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 0\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
\end{align*}\)