Non-classical linear divisibility sequences and cyclotomic polynomials

TL;DR

This paper characterizes non-classical linear divisibility sequences using cyclotomic polynomials, introduces new sequences, and explores their properties, including the lack of strong divisibility unlike Fibonacci numbers.

Contribution

It provides a complete characterization of such sequences via cyclotomic polynomial factorizations and constructs new integer divisibility sequences.

Findings

Sequences are characterized by cyclotomic polynomial factorizations

Constructed new integer divisibility sequences based on these characterizations

Found that these sequences lack strong divisibility property

Abstract

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials that divide their compositions with every positive integer power. We completely characterize such polynomials in terms of their factorizations into cyclotomic polynomials using labeled Hasse diagrams, and construct new integer divisibility sequences based on them. We also show that, unlike the Fibonacci numbers, these non-classical sequences do not have the property of strong divisibility.