Av(1243, 1324, 1432, 2413, 3142)
Generating Function
\(\displaystyle -\frac{\left(-3 x^{3}+3 x^{2}+\left(x -1\right)^{3} \sqrt{-4 x +1}-3 x +1\right) \left(x -1\right)^{3}}{2 x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 184, 587, 1921, 6426, 21886, 75667, 264956, 937922, 3351336, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1\right) F \left(x
\right)^{2}-\left(3 x^{3}-3 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 184\)
\(\displaystyle a \! \left(7\right) = 587\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(40+21 n \right) a \! \left(1+n \right)}{9+n}-\frac{23 \left(7+3 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(227+66 n \right) a \! \left(n +3\right)}{9+n}-\frac{17 \left(40+9 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(613+111 n \right) a \! \left(n +5\right)}{9+n}-\frac{16 \left(20+3 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 184\)
\(\displaystyle a \! \left(7\right) = 587\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(40+21 n \right) a \! \left(1+n \right)}{9+n}-\frac{23 \left(7+3 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(227+66 n \right) a \! \left(n +3\right)}{9+n}-\frac{17 \left(40+9 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(613+111 n \right) a \! \left(n +5\right)}{9+n}-\frac{16 \left(20+3 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(86+11 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{18} \left(x \right)^{2} F_{0}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)