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###### Av(132, 213)

Generating Function

\(\displaystyle \frac{x -1}{2 x -1}\)

Counting Sequence

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...

Implicit Equation for the Generating Function

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

Recurrence

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

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

\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right), \quad n \geq 2\)

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

\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right), \quad n \geq 2\)

Explicit Closed Form

\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 2^{-1+n} & \text{otherwise} \end{array}\right.\)

### This specification was found using the strategy pack "Point Placements" and has 12 rules.

Found on January 17, 2022.Finding the specification took 0 seconds.

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