On Arithmetic Progressions in Recurrences - A new characterization of the Fibonacci sequence

TL;DR

This paper proves that the Fibonacci sequence is uniquely characterized among binary recurrences by containing infinitely many three-term arithmetic progressions, and provides a criterion for other linear recurrences to have this property.

Contribution

It introduces a new characterization of the Fibonacci sequence based on the presence of infinitely many three-term arithmetic progressions and offers a criterion for general linear recurrences.

Findings

Fibonacci sequence uniquely contains infinitely many three-term arithmetic progressions among binary recurrences.

A criterion for linear recurrences to have infinitely many three-term arithmetic progressions.

The result distinguishes Fibonacci from other similar recurrence sequences.

Abstract

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given.