Av(1324, 1432, 2143)
Generating Function
\(\displaystyle \frac{-x \left(x^{2}-x +1\right)^{2} \sqrt{1-4 x}+3 x^{5}-14 x^{4}+27 x^{3}-26 x^{2}+13 x -2}{2 x^{6}+4 x^{5}-22 x^{4}+38 x^{3}-32 x^{2}+14 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 289, 1103, 4261, 16603, 65100, 256466, 1014107, 4021836, 15988827, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}+2 x^{5}-11 x^{4}+19 x^{3}-16 x^{2}+7 x -1\right) F \left(x
\right)^{2}+\left(-3 x^{5}+14 x^{4}-27 x^{3}+26 x^{2}-13 x +2\right) F \! \left(x \right)+x^{5}-4 x^{4}+9 x^{3}-10 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 289\)
\(\displaystyle a \! \left(7\right) = 1103\)
\(\displaystyle a \! \left(8\right) = 4261\)
\(\displaystyle a \! \left(n +9\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(29+4 n \right) a \! \left(8+n \right)}{8+n}-\frac{3 \left(n -4\right) a \! \left(n +1\right)}{8+n}+\frac{\left(23+49 n \right) a \! \left(n +2\right)}{8+n}-\frac{4 \left(67+35 n \right) a \! \left(n +3\right)}{8+n}+\frac{9 \left(77+24 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(947+214 n \right) a \! \left(n +5\right)}{8+n}+\frac{6 \left(127+23 n \right) a \! \left(n +6\right)}{8+n}-\frac{\left(361+56 n \right) a \! \left(n +7\right)}{8+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 289\)
\(\displaystyle a \! \left(7\right) = 1103\)
\(\displaystyle a \! \left(8\right) = 4261\)
\(\displaystyle a \! \left(n +9\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{3 \left(29+4 n \right) a \! \left(8+n \right)}{8+n}-\frac{3 \left(n -4\right) a \! \left(n +1\right)}{8+n}+\frac{\left(23+49 n \right) a \! \left(n +2\right)}{8+n}-\frac{4 \left(67+35 n \right) a \! \left(n +3\right)}{8+n}+\frac{9 \left(77+24 n \right) a \! \left(n +4\right)}{8+n}-\frac{\left(947+214 n \right) a \! \left(n +5\right)}{8+n}+\frac{6 \left(127+23 n \right) a \! \left(n +6\right)}{8+n}-\frac{\left(361+56 n \right) a \! \left(n +7\right)}{8+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements Req Corrob Expand Verified" and has 23 rules.
Found on January 21, 2022.Finding the specification took 7 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22} \left(x \right)^{2} F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 27 rules.
Found on January 21, 2022.Finding the specification took 14 seconds.
Copy 27 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_{15}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= \frac{F_{10}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{10}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{12}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{16}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{15}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{15}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Expand Verified" and has 25 rules.
Found on January 21, 2022.Finding the specification took 7 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{13}\! \left(x \right) F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{13}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)