I have a function whose time complexity is given by the following recurrence:

\begin{equation*} T(n) = \begin{cases} \mathcal{O}(1) & \text{for } n=0\\ T(k)+T(k-1)+\mathcal{O}(1) & \text{for } n=2k\\ T(k)+\mathcal{O}(1) & \text{for } n=2k+1\\ \end{cases} \end{equation*}

and I have to prove that $$T(n)\in \mathcal{O}(\phi^{\log_2 n}),$$

where $\phi$ is the golden ratio, $(1 + \sqrt5)\over2$.

I think I could prove it by induction but, how would I go on about it if I didn't know that $T(n)\in \mathcal{O}(\phi^{\log_2 n})$ in the first place?

  • $\begingroup$ is $\phi$ here the golden ratio (which is constant)? $\endgroup$ – nir shahar May 29 at 22:19

Your sequence (shifted by 1) is known as Stern's diatomic sequence, or the Stern–Brocot sequences. The usual recurrence is: \begin{align} &a(0)=0 \\ &a(1)=1 \\ &a(2n) = a(n) \\ &a(2n+1) = a(n) + a(n+1) \end{align} The recurrence suggests that the answer has something to do with binary representation, so one might be prompted to look at the maximal value of $a(n)$ among numbers of length $m$ in binary: $$ 1,2,3,5,8,13,\ldots $$ This is the Fibonacci sequence. (One can check that the first maxima are attained at $(2^n-(-1)^n)/3$.)

From here, one immediately sees that the rate of growth is $O(\phi^{\log_2 n})$.

More can be said. For example, Coons and Tyler determined the best possible constant in front of $\phi^{\log_2 n}$ in their paper The maximal order of Stern's diatomic sequence.

You can find many more links in the OEIS entry A002487.

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