Av(1243, 1342, 1423, 2143, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{3}-2 x^{2}+2 x -1\right) \sqrt{1-4 x}+2 x^{5}+2 x^{4}-3 x^{3}+2 x^{2}-2 x +1}{2 x \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 18, 52, 157, 497, 1631, 5493, 18858, 65711, 231734, 825409, 2964962, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{4} F \left(x
\right)^{2}-\left(2 x^{5}+2 x^{4}-3 x^{3}+2 x^{2}-2 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}-2 x^{7}-2 x^{5}+5 x^{4}-7 x^{3}+7 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 157\)
\(\displaystyle a \! \left(7\right) = 497\)
\(\displaystyle a \! \left(8\right) = 1631\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(5+9 n \right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(12+5 n \right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}-\frac{3 \left(1+n \right)}{5+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 157\)
\(\displaystyle a \! \left(7\right) = 497\)
\(\displaystyle a \! \left(8\right) = 1631\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(5+9 n \right) a \! \left(1+n \right)}{5+n}-\frac{2 \left(12+5 n \right) a \! \left(n +2\right)}{5+n}+\frac{2 \left(11+3 n \right) a \! \left(n +3\right)}{5+n}-\frac{3 \left(1+n \right)}{5+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
\end{align*}\)