# Beatty sequences

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.