Number of ways of writing an integer as a sum of other integers

Question

Fix a non-negative integer $m$. For any integer $A \geq 1$, use $P_m(A)$ to denote the number of ways of rewriting $A = A_1 + A_2 + \ldots A_k$, with $ 1 \leq A_i \leq m$, for any $k \geq 1$.

I would like to find an upper bound for $P_m(A)$ as $A$ is large. Do you know some good reference for this estimate?

Answer 1

The sequence $P_m(0),P_m(1),P_m(2)\dots $ is known as the $m$-step fibonacci sequence.

The link contains various asymptotic results for the sequence.