Binary Signed-Digit Integers, the Stern Diatomic Sequence and Stern Polynomials

TL;DR

This paper establishes a novel connection between Stern's diatomic sequence and binary signed-digit representations, enabling new insights and results in the analysis of integer representations and their properties.

Contribution

It introduces a new link between Stern's diatomic sequence and BSD representations, allowing transfer of results and development of new theorems in this area.

Findings

A correspondence between BSD representations and Stern's sequence elements.

A weight-distribution theorem for BSD representations.

A recursion for optimal BSD representations and their Hamming weight.

Abstract

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit (BSD) representation of integers is used widely in efficient computation, coding theory and other applications. We link these two objects, showing that the number of $i$-bit binary signed-digit representations of an integer $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known about the Stern diatomic sequence available for consideration in the study of binary signed-digit integers, and vice versa. Applications of this relationship discussed in this paper include a weight-distribution theorem for BSD representations, linking these representations to Stern polynomials, a recursion for the number of optimal BSD representations of an integer along with their Hamming weight, stemming from an easy recursion for the leading coefficients and degrees of Stern polynomials, and the identification of all integers having a maximal number of such representations.