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
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.
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