Counting non-standard binary representations

Anders, Katie

Journal of integer sequences, Tome 19 (2016) no. 3

Résumé

Let $\mathcal{A}$ be a finite subset of $\mathbb{N} $ including 0 and let $f_\mathcal{A}(n)$ 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 $s_{\mathcal{A}}(r,m)\sim c(\mathcal{A},m)\left\vert\mathcal{A}\right\vert^r$ for some nonzero $c(\mathcal{A},m)\in\mathbb{Q} $.

Détail

Classification : 11A63
Keywords: digital representation, non-standard binary representation, summatory function