Non-standard binary representations and the Stern sequence

TL;DR

This paper establishes a novel connection between the number of short binary signed-digit representations of integers and the Stern sequence, providing multiple proofs and linking to recent related work.

Contribution

It introduces a new combinatorial interpretation of the Stern sequence through binary signed-digit representations and offers various proof techniques for this relationship.

Findings

Number of short binary signed-digit representations equals the Stern sequence

Provides direct, bijective, and generating function proofs

Connects to recent work by Monroe on fixed-length representations

Abstract

We show that the number of short binary signed-digit representations of an integer $n$ is equal to the $n$-th term in the Stern sequence. Various proofs are provided, including direct, bijective, and generating function proofs. We also show that this result can be derived from recent work of Monroe on binary signed-digit representations of a fixed length.