Which relation symbol goes between $F_m$ and $\phi^m$?

Question

I have already asked a similar question where the tilde notation was used (and context can be taken from there). Now I think that tilde is not the correct symbol to go between these two functions since tilde is defined as $\lim\limits_{n\to \infty}\dfrac{f(n)}{g(n)} = 1$, but the functions in question actually tend to $\frac{1}{\sqrt{5}}$, though I might be wrong.

So in terms of asymptotic analysis can anyone help me understand in which relation are these two functions exactly?

Answer 1

The usual approach is to write $F_m\in\Theta(\phi^m)$ or, in an abuse of notation, $F_m=\Theta(\phi^m)$. See here for some other common relations; $\mathcal{O}$ and $\Omega$, in particular, can be accurately used here.