Av(132, 2314)
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Generating Function
\(\displaystyle -\frac{2 x -1}{x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) F \! \left(x \right)+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right), \quad n \geq 2\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(\left(\sqrt{5}-3\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}\right) \left(5+\sqrt{5}\right)}{20}\)
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This specification was found using the strategy pack "Point Placements" and has 17 rules.

Found on January 17, 2022.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)