  Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and
$f_\mathcal{A}(n)$ be the number of ways to write
$n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We
consider asymptotics of the summatory function $s_\mathcal{A}(r,m)$ of
$f_\mathcal{A}(n)$ from $m2^r$ to $m2^{r+1}-1$ and show that
$s_{\mathcal{A}}(r,m)\approx c(\mathcal{A},m)\left|\mathcal{A}\right|^r$ for
some $c(\mathcal{A},m)\in\mathbb{Q}$.
