Suppose R is an irrational number greater than 1, and let S be the number satisfying the equation 1/R + 1/S = 1. Let [x] denote the floor function of x, that is, the greatest integer less than or equal to x. Then the sequences [nR] and [nS], where n ranges through the set N of positive integers, are the Beatty sequences determined by R. The interesting thing about them is that they partition N; in other words, every positive integer occurs exactly once in one sequence or the other. For example, when R is the golden ratio (about 1.618), the two sequences begin with

 

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ..., and

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34....

 

Beatty sequences are named after the American mathematician Samuel Beatty (1881–1970) who introduced them in 1926 in a problem in the American Mathematical Monthly. Beatty was the first person to receive a Ph.D. in mathematics from a Canadian university, a colorful teacher, and a problemist who became the chairman of the mathematics department, and later, Chancellor, of the University of Toronto.