###### Av(123, 132, 231)

Generating Function

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

Counting Sequence

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...

Implicit Equation for the Generating Function

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

Recurrence

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

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

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

\(\displaystyle a \! \left(n \right) = n, \quad n \geq 3\)

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

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

\(\displaystyle a \! \left(n \right) = n, \quad n \geq 3\)

Explicit Closed Form

\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ n & \text{otherwise} \end{array}\right.\)

### This specification was found using the strategy pack "Point Placements" and has 13 rules.

Found on January 17, 2022.Finding the specification took 1 seconds.

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