Av(1243, 1324, 2431, 4132, 4231)
Generating Function
\(\displaystyle \frac{x^{9}-x^{8}-12 x^{7}+44 x^{6}-95 x^{5}+124 x^{4}-95 x^{3}+42 x^{2}-10 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 155, 414, 1087, 2833, 7362, 19112, 49606, 128782, 334482, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{9}-x^{8}-12 x^{7}+44 x^{6}-95 x^{5}+124 x^{4}-95 x^{3}+42 x^{2}-10 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 155\)
\(\displaystyle a \! \left(7\right) = 414\)
\(\displaystyle a \! \left(8\right) = 1087\)
\(\displaystyle a \! \left(9\right) = 2833\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{3}}{6}+\frac{3 n^{2}}{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{4 n}{3}-4, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 155\)
\(\displaystyle a \! \left(7\right) = 414\)
\(\displaystyle a \! \left(8\right) = 1087\)
\(\displaystyle a \! \left(9\right) = 2833\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{3}}{6}+\frac{3 n^{2}}{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{4 n}{3}-4, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n^{3}}{6}+\frac{2^{n} n}{8}-n^{2}-\frac{3 \,2^{n}}{4}+\frac{11 n}{6}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 47 rules.
Found on January 17, 2022.Finding the specification took 6 seconds.
Copy 47 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_{22}\! \left(x \right)+F_{5}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \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_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{41}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
\end{align*}\)