Av(1342, 1423, 2143)
Generating Function
\(\displaystyle \frac{2 x^{2}-2 x +1-\sqrt{-4 x^{5}+16 x^{4}-24 x^{3}+20 x^{2}-8 x +1}}{2 x \left(x^{2}-x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 290, 1118, 4398, 17595, 71385, 293042, 1215035, 5081259, 21408350, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-x +1\right) F \left(x
\right)^{2}+\left(-2 x^{2}+2 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 290\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(17+10 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(59+22 n \right) a \! \left(n +2\right)}{n +8}-\frac{12 \left(18+5 n \right) a \! \left(n +3\right)}{n +8}+\frac{4 \left(59+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(160+29 n \right) a \! \left(n +5\right)}{n +8}+\frac{3 \left(20+3 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 290\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(17+10 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(59+22 n \right) a \! \left(n +2\right)}{n +8}-\frac{12 \left(18+5 n \right) a \! \left(n +3\right)}{n +8}+\frac{4 \left(59+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(160+29 n \right) a \! \left(n +5\right)}{n +8}+\frac{3 \left(20+3 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\
\end{align*}\)