Av(1243, 1342, 2143, 2431, 3142, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{4}-x^{3}+5 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{5}+2 x^{4}+x^{3}-5 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) \left(x -1\right)^{2} x}\)
Counting Sequence
1, 1, 2, 6, 18, 53, 159, 494, 1591, 5283, 17970, 62271, 218926, 778551, 2794638, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{5}-2 x^{4}-x^{3}+5 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}+3 x^{7}-14 x^{6}+2 x^{5}+30 x^{4}-42 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 494\)
\(\displaystyle a \! \left(8\right) = 1591\)
\(\displaystyle a \! \left(9\right) = 5283\)
\(\displaystyle a \! \left(n +7\right) = -\frac{8 \left(-1+2 n \right) a \! \left(n \right)}{8+n}+\frac{4 \left(-9+5 n \right) a \! \left(1+n \right)}{8+n}+\frac{2 \left(83+20 n \right) a \! \left(n +2\right)}{8+n}-\frac{\left(405+107 n \right) a \! \left(n +3\right)}{8+n}+\frac{5 \left(93+20 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(268+47 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(75+11 n \right) a \! \left(n +6\right)}{8+n}-\frac{1}{8+n}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 494\)
\(\displaystyle a \! \left(8\right) = 1591\)
\(\displaystyle a \! \left(9\right) = 5283\)
\(\displaystyle a \! \left(n +7\right) = -\frac{8 \left(-1+2 n \right) a \! \left(n \right)}{8+n}+\frac{4 \left(-9+5 n \right) a \! \left(1+n \right)}{8+n}+\frac{2 \left(83+20 n \right) a \! \left(n +2\right)}{8+n}-\frac{\left(405+107 n \right) a \! \left(n +3\right)}{8+n}+\frac{5 \left(93+20 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(268+47 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(75+11 n \right) a \! \left(n +6\right)}{8+n}-\frac{1}{8+n}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{3}\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)