International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 525-528

Generalized Beatty sequences

A. McD. Mercer

Department of Mathematics and Statistics, University of Guelph, Ontario, Guelph NIG 2WI, Canada

Received 12 April 1978

Copyright © 1978 A. McD. Mercer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A well-known result due to S. Beatty is that if α and β are positive irrational numbers satisfying α1+β1=1 then each positive integer is to be found in precisely one of the sequences {[kα]}, {[kβ]}(k=1,2,3,) where [x] denotes the integral part of x. The present note generalizes this result to the case of the pair of sequences {[f(k)]}, {[g(k)]} with suitable hypotheses on the functions f and g. The special case f(x)=αx, g(x)=βx is the result due to Beatty.