Av(1342, 1423, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{6}-2 x^{5}+4 x^{4}-2 x^{3}-2 x^{2}+3 x -1}{2 x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 186, 602, 1996, 6752, 23221, 80962, 285541, 1016912, 3651938, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(2 x^{6}-2 x^{5}+4 x^{4}-2 x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{7}+x^{5}-x^{4}-3 x^{3}+6 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 186\)
\(\displaystyle a \! \left(7\right) = 602\)
\(\displaystyle a \! \left(8\right) = 1996\)
\(\displaystyle a \! \left(9\right) = 6752\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(5+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(42+13 n \right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(103+29 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(320+77 n \right) a \! \left(n +3\right)}{n +8}+\frac{64 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(95+16 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(62+9 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n +6}{n +8}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 186\)
\(\displaystyle a \! \left(7\right) = 602\)
\(\displaystyle a \! \left(8\right) = 1996\)
\(\displaystyle a \! \left(9\right) = 6752\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(5+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(42+13 n \right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(103+29 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(320+77 n \right) a \! \left(n +3\right)}{n +8}+\frac{64 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(95+16 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(62+9 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n +6}{n +8}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
\end{align*}\)