Av(1243, 1342, 1423, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{2}-x +1\right) \sqrt{1-4 x}+2 x^{6}+2 x^{4}-2 x^{3}+x^{2}+x -1}{2 x \left(x^{4}+2 x^{2}-x +1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 18, 54, 169, 547, 1816, 6154, 21203, 74050, 261552, 932696, 3353339, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{4}+2 x^{2}-x +1\right) \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(2 x^{6}+2 x^{4}-2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{7}-x^{4}+x^{3}+x^{2}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 169\)
\(\displaystyle a \! \left(7\right) = 547\)
\(\displaystyle a \! \left(8\right) = 1816\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(7+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(41+9 n \right) a \! \left(1+n \right)}{9+n}-\frac{2 \left(46+9 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(71+14 n \right) a \! \left(n +3\right)}{9+n}-\frac{2 \left(96+17 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(197+31 n \right) a \! \left(n +5\right)}{9+n}-\frac{2 \left(64+9 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(55+7 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 169\)
\(\displaystyle a \! \left(7\right) = 547\)
\(\displaystyle a \! \left(8\right) = 1816\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(7+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(41+9 n \right) a \! \left(1+n \right)}{9+n}-\frac{2 \left(46+9 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(71+14 n \right) a \! \left(n +3\right)}{9+n}-\frac{2 \left(96+17 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(197+31 n \right) a \! \left(n +5\right)}{9+n}-\frac{2 \left(64+9 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(55+7 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 18 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 18 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
\end{align*}\)