Av(1243, 1324, 1342, 2134, 2341, 4123)
Generating Function
\(\displaystyle \frac{-\left(x^{2}+x -1\right) \left(x -1\right)^{3} \sqrt{1-4 x}-2 x^{9}+2 x^{8}+2 x^{7}-6 x^{6}+x^{5}-x^{3}+5 x^{2}-4 x +1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 18, 50, 146, 452, 1466, 4917, 16879, 58911, 208201, 743188, 2674883, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(x^{2}+x -1\right) \left(2 x^{9}-2 x^{8}-2 x^{7}+6 x^{6}-x^{5}+x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{17}-2 x^{16}-x^{15}+8 x^{14}-6 x^{13}-5 x^{12}+11 x^{11}-8 x^{10}+4 x^{9}+6 x^{8}-5 x^{7}-15 x^{6}+10 x^{5}+25 x^{4}-41 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 146\)
\(\displaystyle a \! \left(7\right) = 452\)
\(\displaystyle a \! \left(8\right) = 1466\)
\(\displaystyle a \! \left(9\right) = 4917\)
\(\displaystyle a \! \left(10\right) = 16879\)
\(\displaystyle a \! \left(11\right) = 58911\)
\(\displaystyle a \! \left(12\right) = 208201\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +6}+\frac{\left(10+7 n \right) a \! \left(n +1\right)}{n +6}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{n +6}-\frac{\left(7 n +24\right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{n +6}-\frac{3 n^{3}-35 n^{2}+50 n +52}{2 \left(n +6\right)}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 146\)
\(\displaystyle a \! \left(7\right) = 452\)
\(\displaystyle a \! \left(8\right) = 1466\)
\(\displaystyle a \! \left(9\right) = 4917\)
\(\displaystyle a \! \left(10\right) = 16879\)
\(\displaystyle a \! \left(11\right) = 58911\)
\(\displaystyle a \! \left(12\right) = 208201\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +6}+\frac{\left(10+7 n \right) a \! \left(n +1\right)}{n +6}-\frac{2 \left(3 n +8\right) a \! \left(n +2\right)}{n +6}-\frac{\left(7 n +24\right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(3 n +14\right) a \! \left(n +4\right)}{n +6}-\frac{3 n^{3}-35 n^{2}+50 n +52}{2 \left(n +6\right)}, \quad n \geq 13\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 79 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_{6}\! \left(x \right)\\
F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{2}\! \left(x \right) &= 1\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x\\
F_{6}\! \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_{54}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{3}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{5}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x , 1\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{59}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{49}\! \left(x \right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{64}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= y x\\
F_{68}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{65}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= 2 F_{28}\! \left(x \right)+F_{70}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{67}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{3}\! \left(x \right)+F_{72}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{61}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= \frac{y F_{57}\! \left(x , y\right)-F_{57}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)