Av(13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 23451, 23541, 24153, 24351, 24531, 25143, 25341, 25431, 34152, 34251, 35142, 35241)
Generating Function
\(\displaystyle \frac{4 x^{3}-6 x^{2}+5 x -1}{\left(4 x -1\right) \left(2 x^{2}-2 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 408, 1640, 6560, 26224, 104864, 419424, 1677696, 6710848, 26843520, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) \left(2 x^{2}-2 x +1\right) F \! \left(x \right)-4 x^{3}+6 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 10 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 10 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \left(\frac{1}{5}-\frac{i}{10}\right) \left(\frac{1}{2}-\frac{i}{2}\right)^{-n}+\\\left(\frac{1}{5}+\frac{i}{10}\right) \left(\frac{1}{2}+\frac{i}{2}\right)^{-n}+\frac{4^{n}}{10} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 25 rules.
Finding the specification took 49 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\
\end{align*}\)