###### Av(123, 231)

Generating Function

\(\displaystyle -\frac{2 x^{2}-2 x +1}{\left(x -1\right)^{3}}\)

Counting Sequence

1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, ...

Implicit Equation for the Generating Function

\(\displaystyle \left(x -1\right)^{3} F \! \left(x \right)+2 x^{2}-2 x +1 = 0\)

Recurrence

\(\displaystyle a \! \left(0\right) = 1\)

\(\displaystyle a \! \left(1\right) = 1\)

\(\displaystyle a \! \left(2\right) = 2\)

\(\displaystyle a \! \left(n \right) = -\frac{1}{2} n +\frac{1}{2} n^{2}+1, \quad n \geq 3\)

\(\displaystyle a \! \left(1\right) = 1\)

\(\displaystyle a \! \left(2\right) = 2\)

\(\displaystyle a \! \left(n \right) = -\frac{1}{2} n +\frac{1}{2} n^{2}+1, \quad n \geq 3\)

Explicit Closed Form

\(\displaystyle -\frac{1}{2} n +\frac{1}{2} n^{2}+1\)

### This specification was found using the strategy pack "Row Placements Expand Verified" and has 15 rules.

Found on October 26, 2021.Finding the specification took 0 seconds.

\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)\\
\end{align*}\)