Av(123, 231)
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Generating Function
\(\displaystyle -\frac{2 x^{2}-2 x +1}{\left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{3} F \! \left(x \right)+2 x^{2}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n \right) = -\frac{1}{2} n +\frac{1}{2} n^{2}+1, \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle -\frac{1}{2} n +\frac{1}{2} n^{2}+1\)
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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.

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