Convolution companions to Fibonacci, Pell and Lucas sequences

A mathematical concept (whether familiar or unfamiliar) can enhance student engagement when structured into sequential steps, turning the investigation into a journey of discovery. Our goal in this paper is to structure the learning process into a step-by-step path of inquiry and discovery. The convolution companion (CC) of an infinite sequence connects two active chapters of elementary number theory: Fibonacci numbers and Multiplicative arithmetic functions. The convolution companion and Dirichlet inverse (of a multiplicative function) are two related concepts that complement each other. Properties of linear recurrence sequences are examined using the tools of the theory of multiplicative functions. This approach enables the discovery of new connections within elementary number theory, offering a new framework for systematic and objective inquiry.