Binary Signed-Digit Integers and the Stern Diatomic Sequence

TL;DR

This paper establishes a novel connection between binary signed-digit representations of integers and Stern's diatomic sequence, enabling new insights and applications in computation and cryptography.

Contribution

It reveals that the count of i-bit binary signed-digit representations of an integer relates directly to Stern's diatomic sequence, linking two important mathematical concepts.

Findings

Number of i-bit representations equals a specific Stern sequence element.

Provides a new perspective for analyzing signed-digit representations.

Enables transfer of known results from Stern's sequence to signed-digit analysis.

Abstract

Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient computation, coding theory and cryptography. We link these two ideas here, showing that the number of $i$-bit binary signed-digit representations of an integer $n$ with $n<2^i$ is the $(2^i-n)^\text{th}$ element in Stern's diatomic sequence. This correspondence makes the vast range of results known for Stern's diatomic sequence available for consideration in the study of binary signed-digit integers.