## Beatty sequencesSuppose 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. ## Related category• SERIES AND SEQUENCES | |||||||

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