Av(1324, 1342, 2341, 4231)
Generating Function
\(\displaystyle -\frac{3 x^{5}-15 x^{4}+24 x^{3}-19 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 209, 634, 1855, 5277, 14697, 40293, 109186, 293329, 782996, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+3 x^{5}-15 x^{4}+24 x^{3}-19 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n}{2}+1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{n}{2}+1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{\left(3 \sqrt{5}+5\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-3 \sqrt{5}+5\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{n^{2}}{2}+\frac{n}{2}-3 \,2^{n}+3\)
This specification was found using the strategy pack "Col Placements" and has 104 rules.
Found on July 23, 2021.Finding the specification took 11 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_{5}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \left(x \right) &= 2 F_{82}\! \left(x \right)+F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 2 F_{44}\! \left(x \right)+F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{36}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \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_{5}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 0\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{27}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{20}\! \left(x \right) F_{30}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{20}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{1}\! \left(x \right)+F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{5}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{5}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{5}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{5}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{5}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{28}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{5}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{5}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{5}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{30}\! \left(x \right) F_{5}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{5}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{5}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{103}\! \left(x \right)+F_{78}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{5}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{87}\! \left(x \right)+F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{5}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= 2 F_{89}\! \left(x \right)+F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{5}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{5}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{5}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{20}\! \left(x \right) F_{5}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{100}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{5}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{5}\! \left(x \right) F_{84}\! \left(x \right)\\
\end{align*}\)