Beatty Sequences, Continued Fractions, and Certain Shift Operators

Abstract

Let θ = θ ( k ) be the positive root of θ 2 + ( k -2) θ - k = 0. Let f ( n ) = [( n + l) θ ]-[n θ ] for positive integers n , where [ x ] denotes the greatest integer in x . Then the elements of the infinite sequence ( f (l), f (2), f (3),…) can be rapidly generated from the finite sequence ( f (l), f (2),…, f ( k )) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [ n θ ] itself in this manner.