Av(1243, 1324, 3412)
Generating Function
\(\displaystyle -\frac{10 x^{9}-64 x^{8}+192 x^{7}-367 x^{6}+447 x^{5}-358 x^{4}+189 x^{3}-63 x^{2}+12 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 237, 711, 1988, 5253, 13301, 32673, 78669, 187230, 443398, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+10 x^{9}-64 x^{8}+192 x^{7}-367 x^{6}+447 x^{5}-358 x^{4}+189 x^{3}-63 x^{2}+12 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 237\)
\(\displaystyle a \! \left(7\right) = 711\)
\(\displaystyle a \! \left(8\right) = 1988\)
\(\displaystyle a \! \left(9\right) = 5253\)
\(\displaystyle a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{\left(n -1\right) \left(3 n^{4}-12 n^{3}-97 n^{2}-22 n -120\right)}{120}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 237\)
\(\displaystyle a \! \left(7\right) = 711\)
\(\displaystyle a \! \left(8\right) = 1988\)
\(\displaystyle a \! \left(9\right) = 5253\)
\(\displaystyle a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)+\frac{\left(n -1\right) \left(3 n^{4}-12 n^{3}-97 n^{2}-22 n -120\right)}{120}, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}-\frac{n^{5}}{40}-\frac{13 n^{3}}{24}-n^{2}+2^{n} n -\frac{73 n}{30}+2^{n +1}-2\)
This specification was found using the strategy pack "Insertion Point Row Placements" and has 106 rules.
Found on July 23, 2021.Finding the specification took 15 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_{13}\! \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_{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_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
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_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{13}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \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_{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_{13}\! \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_{13}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{13}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{46}\! \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_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{13}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{21}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{59}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{31} \left(x \right)^{2} F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{13}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{13}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{72}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{13}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{13}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{13}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{13}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{13}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{13}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{13}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{13}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{25}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{13}\! \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_{24}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{11}\! \left(x \right) F_{31}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{25}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{21}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{31}\! \left(x \right) F_{93}\! \left(x \right)\\
\end{align*}\)