Av(123, 132)
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Generating Function
\(\displaystyle \frac{x -1}{2 x -1}\)
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Counting Sequence
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) F \! \left(x \right)+1-x = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right), \quad n \geq 2\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 2^{-1+n} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 15 rules.

Found on January 17, 2022.

Finding the specification took 1 seconds.

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