Journal of Integer Sequences, Vol. 19 (2016), Article 16.3.3

Counting Non-Standard Binary Representations


Katie Anders
Department of Mathematics
University of Texas at Tyler
3900 University Blvd.
Tyler, TX 75799
USA

Abstract:

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 m2r to m2r+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} $.

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Received August 25 2015; revised versions received January 19 2016; March 11 2016; April 5 2016. Published in Journal of Integer Sequences, April 6 2016.

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