Descriptions of the Characteristic Sequence of an Irrational

Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is f (α) = f1 f2 ; where fn = [(n+1)α] [nα]: We make some observations on the various descriptions of the characteristic sequence of α which have appeared in the literature. We then refine one of these descriptions in order to obtain a very simple derivation of an arithmetic expression for [nα] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoini [17]. Some concluding remarks give conditions on n which are equivalent to fn = 1.