A graphical approach to the Fibonacci sequence (Fn) n≥0 modulo m extended to the Lucas sequences (Un(p,q))n≥0 and (Vn(P,q))n≥0

aBa Mbirika & Morgan Fiebig
University of Wisconsin – Eau Claire

The goal of this talk is twofold: (1) extend theory on statistics in the Fibonacci and Lucas sequences modulo m to the Lucas sequences U :=(Un(p,q))n≥0 and V :=(Vn(p,q)n 0, and (2) apply this theory to a novel graphical approach of U and V modulo m. The statistics we explore are the period π(m), entry point e(m), and order ω(m) := pi(m)/e(m). We generalize a wealth of known Fibonacci and Lucas statistical results to the U and V setting. Based on ω(m), we describe behaviors shared by infinite families of nondegenerate U and V sequences with parameters q = ± 1. In our graphical approach we place the cycle of repeating terms of the periods of U and V in a circle, and patterns which would otherwise be overlooked emerge. In particular, we exhibit some tantalizing examples in the following three sequence pairs: Fibonacci and Lucas, Pell and associated Pell, and, balancing and Lucas-balancing. Our proofs utilize results from primary sources ranging from the ground-breaking papers of Lucas in 1878 and Carmichael in 1913, to the seminal works of Wall in 1960 and Vinson in 1963, amongst others.