Period patterns, entry points, and orders in the Lucas sequences: theory and applications

The goal of this paper is twofold: (1) extend theory on certain statistics in the Fibonacci and Lucas sequences modulo $m$ to the Lucas sequences $U := \left(U_n(p,q)\right)_{n \geq 0}$ and $V := \left(V_n(p,q)\right)_{n \geq 0}$, and (2) apply some of this theory to a novel graphical approach of $U$ and $V$ modulo $m$. Upon placing the cycle of repeating sequence terms in a circle, several fascinating patterns which would otherwise be overlooked emerge. We generalize a wealth of known Fibonacci and Lucas statistical identities to the $U$ and $V$ settings using primary sources such as Lucas in 1878, Carmichael in 1913, Wall in 1960, and Vinson in 1963, amongst others. We use many of these generalized identities to form the theoretical basis for our graphical results. Based on the order of $m$, defined as $\omega(m) := \frac{\pi(m)}{e(m)}$, where $\pi(m)$ is the period of $m$ and $e(m)$ is the entry point of $m$, we describe behaviors shared by $U$ and $V$ with parameters $q = \pm 1$. 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.