Av(1324, 2143, 3412)
Generating Function
\(\displaystyle -\frac{10 x^{6}-38 x^{5}+64 x^{4}-62 x^{3}+33 x^{2}-9 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 21, 71, 222, 648, 1797, 4807, 12548, 32236, 82009, 207529, 524060, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+10 x^{6}-38 x^{5}+64 x^{4}-62 x^{3}+33 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 71\)
\(\displaystyle a \! \left(6\right) = 222\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{5 n}{2}+1, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 71\)
\(\displaystyle a \! \left(6\right) = 222\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{5 n}{2}+1, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{\left(-2 \sqrt{5}+10\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(2 \sqrt{5}+10\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(10 n -60\right) 2^{n}}{10}+\frac{n^{2}}{2}+\frac{7 n}{2}+5\)
This specification was found using the strategy pack "Insertion Point Placements" and has 88 rules.
Found on July 23, 2021.Finding the specification took 12 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 88 equations to clipboard:
\(\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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \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_{8}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 0\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{45}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{14} \left(x \right)^{3}\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{20}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{13}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{13}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{24}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{24}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{33}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{67}\! \left(x \right)\\
\end{align*}\)