Enumerating the rationals from left to right

TL;DR

This paper explores the generation of rational number sequences like Farey, Stern-Brocot, and Calkin-Wilf using similar recurrence relations, highlighting their applications across various fields.

Contribution

It reveals the common recurrence relations underlying these rational sequences and discusses their diverse applications.

Findings

Sequences are generated by nearly identical second order recurrence relations.

These sequences have significant combinatorial, computational, and geometric applications.

The paper provides a unified perspective on enumerating rational numbers.

Abstract

Farey sequences, Stern-Brocot sequences, the Calkin-Wilf sequences are shown to be generated via almost identical second order recurrence relations. These sequences have combinatorial, computational, and geometric applications, and are useful for enumerating the rational numbers.