Recursions, Trains, Trees, and Combinatorial Rod Set Algebra

TL;DR

This paper introduces an algebraic framework based on rod trains and trees to analyze recursive sequences, proving identities, divisibility properties, and polynomial factorizations with numerous examples.

Contribution

It develops a novel algebraic model connecting physical rod trains and tree structures to study recursive sequences and their properties.

Findings

Proves classic identities for recursive sequences.

Shows Lucas sequences are divisibility sequences.

Identifies cyclotomic polynomial factors of Borwein trinomials.

Abstract

We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the subtrees define algebraic operations on sets of rods. We use this algebra to prove classic identities for recursively defined sequences, to show that Lucas sequences are divisibility sequences, to characterize two-term linear Fibonacci identities, and to find the cyclotomic polynomial factors of Borwein trinomials. We complement abstractions with lots of examples.