  Recently, a new model of quantum walk, utilizing recycled coins, was
introduced; however little is yet known about its properties. In this paper, we
study its behavior on the cycle graph. In particular, we will consider its time
averaged distribution and how it is affected by the walk's "memory parameter" -
a real parameter, between zero and eight, which affects the walk's coin flip
operator. Despite an infinite number of different parameters, our analysis
provides evidence that only a few produce non-uniform behavior. Our analysis
also shows that the initial state, and cycle size modulo four all affect the
behavior of this walk. We also prove an interesting relationship between the
recycled coin model and a different memory-based quantum walk recently
proposed.
