On the general solution of the Heideman-Hogan family of recurrences

TL;DR

This paper explores the general solutions of a family of nonlinear rational recurrences, demonstrating their Laurent property and integrality, and extending previous results to the entire solution space.

Contribution

It provides a comprehensive analysis of the general solutions of Heideman-Hogan recurrences, including their Laurent property and integrality for all initial conditions.

Findings

All recurrences have the Laurent property.

Sequences with initial data all ones are integer sequences.

General solutions satisfy linear recurrence relations.

Abstract

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan. All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.