Av(1243, 1342, 1423, 2134, 2314, 2341, 4123)
Generating Function
\(\displaystyle \frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}-2 x^{6}+2 x^{5}+2 x^{4}+x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 17, 46, 137, 435, 1437, 4870, 16805, 58796, 208023, 742912, 2674453, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x^{6}-2 x^{5}-2 x^{4}-x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{11}-2 x^{10}-x^{9}+2 x^{8}+3 x^{6}-2 x^{5}-3 x^{3}+6 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 46\)
\(\displaystyle a \! \left(6\right) = 137\)
\(\displaystyle a \! \left(7\right) = 435\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{2}-4 n -2}{n +2}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 46\)
\(\displaystyle a \! \left(6\right) = 137\)
\(\displaystyle a \! \left(7\right) = 435\)
\(\displaystyle a \! \left(1+n \right) = \frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +2}-\frac{3 n^{2}-4 n -2}{n +2}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 40 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 40 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_{25}\! \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_{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_{21}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{34}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= y x\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{39}\! \left(x , y\right) &= \frac{y F_{28}\! \left(x , y\right)-F_{28}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)