Let $\mathbb{N}=\{1,2,3,\ldots\}$. Find a closed form or an asymptotic form of $f: \mathbb{N} \to \mathbb{N}$, where $f$ satisfies $f(1) = 1$ and $$f(n) = f(n-1) + f\bigg(\left\lfloor \frac{n}{2} \right\rfloor\bigg)\,.$$
What is a closed-form solution to this recurrence relation?
Neither the Master theorem, nor its generalization, the Akra-Bazzi method, seem to be applicable.