Congruence properties of binary partition functions

TL;DR

This paper proves that for a finite set A of natural numbers including zero, the sequence counting binary representations modulo 2 is periodic, with a computably determined period T, and discusses related variations.

Contribution

It establishes the existence of a computable period T for the binary partition function modulo 2, extending understanding of its congruence properties.

Findings

The sequence (f(n) mod 2) is periodic.

The period T is computable from A.

The results generalize to variations of the problem.

Abstract

Let A be a finite subset of the natural numbers containing 0, and let f(n) denote the number of ways to write n in the form $\sum e_j2^j$, where $\e_j \in A$. We show that there exists a computable T = T(A) so that the sequence (f(n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.