Av(1243, 1342, 3124)
Generating Function
\(\displaystyle \frac{\left(-6 x^{4}+12 x^{3}-11 x^{2}+5 x -1\right) \sqrt{1-4 x}-4 x^{4}+14 x^{3}-15 x^{2}+7 x -1}{18 x^{5}-52 x^{4}+68 x^{3}-46 x^{2}+16 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 277, 1016, 3756, 13994, 52491, 197987, 750185, 2853359, 10888249, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(9 x^{5}-26 x^{4}+34 x^{3}-23 x^{2}+8 x -1\right) F \left(x
\right)^{2}+\left(4 x^{4}-14 x^{3}+15 x^{2}-7 x +1\right) F \! \left(x \right)+x^{2} \left(4 x^{2}-5 x +2\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 277\)
\(\displaystyle a \! \left(7\right) = 1016\)
\(\displaystyle a \! \left(8\right) = 3756\)
\(\displaystyle a \! \left(9\right) = 13994\)
\(\displaystyle a \! \left(n +10\right) = -\frac{108 \left(1+2 n \right) a \! \left(n \right)}{n +10}+\frac{30 \left(50+37 n \right) a \! \left(n +1\right)}{n +10}-\frac{6 \left(1009+454 n \right) a \! \left(n +2\right)}{n +10}+\frac{\left(12924+4123 n \right) a \! \left(n +3\right)}{n +10}-\frac{\left(17258+4225 n \right) a \! \left(n +4\right)}{n +10}+\frac{\left(15404+3041 n \right) a \! \left(n +5\right)}{n +10}-\frac{17 \left(552+91 n \right) a \! \left(n +6\right)}{n +10}+\frac{\left(3856+545 n \right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(509+63 n \right) a \! \left(n +8\right)}{n +10}+\frac{\left(154+17 n \right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 277\)
\(\displaystyle a \! \left(7\right) = 1016\)
\(\displaystyle a \! \left(8\right) = 3756\)
\(\displaystyle a \! \left(9\right) = 13994\)
\(\displaystyle a \! \left(n +10\right) = -\frac{108 \left(1+2 n \right) a \! \left(n \right)}{n +10}+\frac{30 \left(50+37 n \right) a \! \left(n +1\right)}{n +10}-\frac{6 \left(1009+454 n \right) a \! \left(n +2\right)}{n +10}+\frac{\left(12924+4123 n \right) a \! \left(n +3\right)}{n +10}-\frac{\left(17258+4225 n \right) a \! \left(n +4\right)}{n +10}+\frac{\left(15404+3041 n \right) a \! \left(n +5\right)}{n +10}-\frac{17 \left(552+91 n \right) a \! \left(n +6\right)}{n +10}+\frac{\left(3856+545 n \right) a \! \left(n +7\right)}{n +10}-\frac{2 \left(509+63 n \right) a \! \left(n +8\right)}{n +10}+\frac{\left(154+17 n \right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 30 rules.
Found on July 23, 2021.Finding the specification took 20 seconds.
Copy 30 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= y x\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{23}\! \left(x \right) F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\
F_{27}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)