Beatty sequence ( $⌊ n α ⌋$ )

⌊ n α ⌋

for

n = 1, 2, \dots

, with a fixed positive irrational number

α

. Complementary Beatty sequences for

α, β > 1

partition the positive integers when

1/ α + 1/ β = 1

Beatty sequence reciprocity

α, β > 1

1/ α + 1/ β = 1

⌊ n α ⌋

and

⌊ n β ⌋

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The Beatty sequence is a sequence of numbers that can be derived from the mathematical concept of filling the real line with two sequences whose terms are the floor functions of the multiples of two irrational numbers.