The properties of linear, homogeneous, constant coefficient, second order recurrence relations such as

are well known. Consider however the recurrence relation

Such 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 x 0, x 1, the sequence x 0, x 1, x 2, x 3, … satisfies x n+9 = x n 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.