Patterns in the Pisano Period and Entry Points of Linear Recurrence Sequences Modulo M

Abstract

The Pisano periods and entry points in the Fibonacci sequence has been a well-studied subject since D.D. Wall’s seminal paper in 1960. The Pisano period is the length of consecutive sequence terms which repeat infinitely to comprise the entire sequence modulo m, the entry point is the index at which the first zero appears in the sequence modulo m, and the order of m is the ratio of the Pisano period to the entry point. We have extended this study to other well known second-order linear recurrence sequences modulo m: Lucas, Pell, associated Pell, balancing, Lucas-balancing, cobalancing, and Lucas-cobalancing. Using data collected through programming in Mathematica, we have observed that the order of m can be used to predict many interesting patterns in the sequence terms modulo m. In particular, we observe that cyclic group structures, palindromes, and semi-palindromic behaviors can all be predicted based on the order of m, and that the m values for which the order of m is four form multiplicative monoids in some settings.