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.