Odd behavior in the coefficients of reciprocals of binary power series

Abstract

Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_{\mathcal{A}}(n)$ be the number of ways to write $n={\sum }_{i=0}^{\infty }{\epsilon }_i{2}^i$, where ${\epsilon }_i\in \mathcal{A}$. The sequence $\left(f_{\mathcal{A}}(n)\right)\mathrm{mod}2$ is always periodic, and $f_{\mathcal{A}}(n)$ is typically more often even than odd. We give four families of sets $\left({\mathcal{A}}_m\right)$ with $\left|{\mathcal{A}}_m\right|=4$ such that the proportion of odd $f_{{\mathcal{A}}_m}(n)$'s goes to $1$ as $m\to \infty$.