PermPal Database

Generating Function

- \frac{{(x - 1)}^{2}}{x^{3} - 2 x^{2} + 3 x - 1}

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Counting Sequence

1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, ...

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Implicit Equation for the Generating Function

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

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

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

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Explicit Closed Form

- \frac{2 (\sum_{α = RootOf (Z^{3} - 2 Z^{2} + 3 Z - 1)} α^{- n + 1})}{23} + \frac{8 (\sum_{α = RootOf (Z^{3} - 2 Z^{2} + 3 Z - 1)} α^{- n})}{23} + \frac{(\sum_{α = RootOf (Z^{3} - 2 Z^{2} + 3 Z - 1)} α^{- n - 1})}{23}

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Found on January 18, 2022.

Finding the specification took 3 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_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{10} (x) + F_{6} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{9} (x) \\ F_{9} (x) & = F_{1} (x) + F_{4} (x) \\ F_{10} (x) & = F_{11} (x) + F_{2} (x) \\ F_{11} (x) & = F_{12} (x) + F_{13} (x) + F_{29} (x) \\ F_{12} (x) & = 0 \\ F_{13} (x) & = F_{14} (x) F_{4} (x) \\ F_{14} (x) & = F_{15} (x) + F_{19} (x) \\ F_{15} (x) & = F_{16} (x) + F_{7} (x) \\ F_{16} (x) & = F_{17} (x) \\ F_{17} (x) & = F_{18} (x) F_{4} (x) \\ F_{18} (x) & = F_{4} (x) \\ F_{19} (x) & = F_{11} (x) + F_{20} (x) \\ F_{20} (x) & = 2 F_{12} (x) + F_{21} (x) + F_{25} (x) \\ F_{21} (x) & = F_{22} (x) F_{4} (x) \\ F_{22} (x) & = F_{23} (x) + F_{24} (x) \\ F_{23} (x) & = F_{16} (x) \\ F_{24} (x) & = F_{20} (x) \\ F_{25} (x) & = F_{26} (x) F_{4} (x) \\ F_{26} (x) & = F_{27} (x) \\ F_{27} (x) & = F_{28} (x) \\ F_{28} (x) & = F_{2} (x) F_{4} (x) \\ F_{29} (x) & = F_{30} (x) F_{4} (x) \\ F_{30} (x) & = F_{2} (x) + F_{27} (x) \end{aligned}

Proof Tree