Av(123, 132, 213)
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Generating Function
\(\displaystyle -\frac{1}{x^{2}+x -1}\)
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Counting Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = a \! \left(n \right)+a \! \left(n +1\right), \quad n \geq 2\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(5+\sqrt{5}\right) \left(\left(\sqrt{5}-3\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}\right)}{20}\)
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This specification was found using the strategy pack "Point Placements" and has 10 rules.

Found on January 17, 2022.

Finding the specification took 0 seconds.

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