# Re$^3$counting the rationals

## TL;DR

This paper explores multiple methods to enumerate positive rationals, introduces new constructions, and generalizes these using circle packings, revealing fundamental limitations and intriguing sequence properties.

## Contribution

It presents several novel and existing enumeration techniques for rationals, extends them via circle packings, and analyzes their mathematical properties and limitations.

## Key findings

- Three enumeration methods are proven to be the only possibilities using circle packings.
- The paper introduces new constructions involving negative continued fractions and Chebyshev polynomials.
- Sequences derived from these enumerations exhibit remarkable similarities and differences.

## Abstract

In 1999, Neil Calkin and Herbert Wilf wrote "Recounting the rationals" which gave an explicit bijection between the positive integers and the positive rationals. We find several different (some new) ways to construct this enumeration and thus create pointers for generalizing. Next, we use circle packings to generalize and find two other enumerations. Surprisingly, the three enumerations are all that are possible by using this technique. The proofs involve, among other things, "negative" continued fractions, Chebyshev polynomials, Euler's totient function, and generalizations of Stern's diatomic sequence. Finally we look at some of the remarkable similarities -- and differences -- of these sequences.