Odd behavior in the coefficients of reciprocals of binary power series

TL;DR

This paper investigates the parity patterns of coefficients in the reciprocals of binary power series, revealing that for certain sets, the proportion of odd coefficients approaches one as the set size increases.

Contribution

It introduces four families of sets with four elements each where the odd coefficient proportion tends to one, highlighting unexpected parity behavior in binary power series reciprocals.

Findings

The sequence of coefficients modulo 2 is always periodic.

Typically, the coefficients are more often even than odd.

For specific set families, the odd coefficient proportion approaches 1.

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 $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. The sequence $\left(f_\mathcal{A}(n)\right) \bmod 2$ 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 $m\to\infty$.