Piecewise linear recurrence relations

Abstract

The properties of linear, homogeneous, constant coefficient, second order recurrence relations such asare well known. Consider however the recurrence relationSuch things are rarely studied; it may not be obvious that equation (2) generates a sequence which is periodic with period 9, which is to say that for any initial values x0, x1, the sequence x0, x1, x2, x3, … satisfies xn+9 = xn for every n. This article records some discoveries which resulted from an attempt to understand this interesting fact. These discoveries open up a field for exciting, elementary, but by no means trivial, investigations.