Av(1324, 1342, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}-3 x^{4}+4 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 177, 554, 1790, 5951, 20241, 70075, 245997, 873285, 3128951, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(3 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-2 x^{7}+7 x^{6}-15 x^{5}+23 x^{4}-24 x^{3}+16 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 177\)
\(\displaystyle a \! \left(7\right) = 554\)
\(\displaystyle a \! \left(8\right) = 1790\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(3+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+9 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(56+17 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}-\frac{2 n}{7+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 177\)
\(\displaystyle a \! \left(7\right) = 554\)
\(\displaystyle a \! \left(8\right) = 1790\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(3+13 n \right) a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+9 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(56+17 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}-\frac{2 n}{7+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)\\
\end{align*}\)