Av(1243, 1324, 3241)
Generating Function
\(\displaystyle -\frac{5 x^{8}-51 x^{7}+172 x^{6}-288 x^{5}+283 x^{4}-170 x^{3}+61 x^{2}-12 x +1}{\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 258, 845, 2649, 8019, 23630, 68216, 193861, 544312, 1514024, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+5 x^{8}-51 x^{7}+172 x^{6}-288 x^{5}+283 x^{4}-170 x^{3}+61 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 258\)
\(\displaystyle a \! \left(7\right) = 845\)
\(\displaystyle a \! \left(8\right) = 2649\)
\(\displaystyle a \! \left(n +6\right) = -4 a \! \left(n \right)+28 a \! \left(n +1\right)-69 a \! \left(n +2\right)+74 a \! \left(n +3\right)-39 a \! \left(n +4\right)+10 a \! \left(n +5\right)+\frac{\left(n +3\right) \left(n -2\right)}{2}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 258\)
\(\displaystyle a \! \left(7\right) = 845\)
\(\displaystyle a \! \left(8\right) = 2649\)
\(\displaystyle a \! \left(n +6\right) = -4 a \! \left(n \right)+28 a \! \left(n +1\right)-69 a \! \left(n +2\right)+74 a \! \left(n +3\right)-39 a \! \left(n +4\right)+10 a \! \left(n +5\right)+\frac{\left(n +3\right) \left(n -2\right)}{2}, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left(-5 n +6\right) \sqrt{5}+15 n +50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(\left(5 n -6\right) \sqrt{5}+15 n +50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{n^{2}}{2}-2^{n -1} n +\frac{n}{2}-2^{n +1}+1\)
This specification was found using the strategy pack "Point Placements" and has 119 rules.
Found on January 18, 2022.Finding the specification took 7 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 119 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_{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_{90}\! \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_{49}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \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_{41}\! \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_{14}\! \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_{41}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{51}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{55}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{51}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{12}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{12}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{31}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{75}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{100}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{105}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{108}\! \left(x \right) &= 2 F_{31}\! \left(x \right)+F_{109}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\
\end{align*}\)