PermPal Database

Generating Function

- \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

(x^{2} - 3 x + 1) F (x) + 2 x - 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (n + 2) = - a (n) + 3 a (n + 1), n \geq 2

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Explicit Closed Form

- \frac{((\sqrt{5} - 3) {(\frac{3}{2} - \frac{\sqrt{5}}{2})}^{- n} - 2 {(\frac{3}{2} + \frac{\sqrt{5}}{2})}^{- n}) (5 + \sqrt{5})}{20}

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Found on January 17, 2022.

Finding the specification took 1 seconds.

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+ Expand

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

Proof Tree