The properties of linear, homogeneous, constant coefficient, second order recurrence relations such as
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170915061659232-0247:S0025557200151629:S0025557200151629_inline1.gif?pub-status=live)
are well known. Consider however the recurrence relation
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20170915061659232-0247:S0025557200151629:S0025557200151629_inline2.gif?pub-status=live)
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 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.