Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 21354, 21453, 21543, 23451, 23541, 24351, 24531, 25341, 25431, 31452, 31542, 32451, 32541, 34521, 35421, 41532, 42531, 43521)
Generating Function
\(\displaystyle \frac{2 x^{6}-2 x^{5}+2 x^{4}-14 x^{3}+17 x^{2}-7 x +1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 315, 1043, 3318, 10248, 30954, 91894, 269093, 779285, 2236234, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-2 x^{6}+2 x^{5}-2 x^{4}+14 x^{3}-17 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 315\)
\(\displaystyle a{\left(n + 6 \right)} = a{\left(n \right)} - 5 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)} + 11 a{\left(n + 3 \right)} - 16 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)} - 1, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 315\)
\(\displaystyle a{\left(n + 6 \right)} = a{\left(n \right)} - 5 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)} + 11 a{\left(n + 3 \right)} - 16 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)} - 1, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{\left(\left(\sqrt{5}\, n -3 n +2\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\sqrt{5}-2 n +3\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\left(\sqrt{5}-3\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\sqrt{5}-2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-5\right) \left(5+\sqrt{5}\right)}{20}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 89 rules.
Finding the specification took 355 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_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{18}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 0\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{18}\! \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_{18}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)+F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{18}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{39}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{18}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{7}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{18}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{18}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 0\\
F_{84}\! \left(x \right) &= F_{18}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{39}\! \left(x \right)+F_{84}\! \left(x \right)\\
\end{align*}\)