Av(1243, 1342, 1423, 1432, 2143, 35142, 354162, 461325, 465132)
Generating Function
\(\displaystyle \frac{\left(3 x^{2}-5 x +1\right) \sqrt{1-4 x}+4 x^{3}-5 x^{2}+5 x -1}{8 x^{2}-2 x}\)
Counting Sequence
1, 1, 2, 6, 19, 63, 216, 759, 2717, 9867, 36244, 134368, 501942, 1886966, 7131840, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x -1\right) F \left(x
\right)^{2}-\left(4 x -1\right) \left(x^{2}-x +1\right) F \! \left(x \right)+x^{4}-4 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = \frac{6 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{\left(30+23 n \right) a \! \left(1+n \right)}{n +4}+\frac{3 \left(8+3 n \right) a \! \left(n +2\right)}{n +4}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = \frac{6 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{\left(30+23 n \right) a \! \left(1+n \right)}{n +4}+\frac{3 \left(8+3 n \right) a \! \left(n +2\right)}{n +4}, \quad n \geq 5\)
This specification was found using the strategy pack "All The Strategies 2 Expand Verified" and has 16 rules.
Found on January 21, 2022.Finding the specification took 41 seconds.
Copy 16 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{8} \left(x \right)^{3}\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Row And Col Placements Expand Verified" and has 32 rules.
Found on January 21, 2022.Finding the specification took 35 seconds.
Copy 32 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_{10}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \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_{10}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{0}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= -F_{11}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{10}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Point Placements Req Corrob Expand Verified" and has 24 rules.
Found on January 21, 2022.Finding the specification took 30 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\
\end{align*}\)