# Definition:Beatty Sequence

## Definition

Let $x$ be an irrational number.

The Beatty sequence on $x$ is the integer sequence $\BB_x$ defined as:

$\BB_x := \sequence{\floor{n x} }_{n \mathop \in \Z_{\ge 0} }$

That is, the terms are the floors of the successive integer multiples of $x$.

### Complementary Beatty Sequence

Let $\mathcal B_x$ be the Beatty sequence on $x$.

The complementary Beatty sequence on $x$ is the integer sequence formed by the integers which are missing from $\mathcal B_x$.

## Also known as

A Beatty sequence is also known as a homogeneous Beatty sequence, to distinguish it specifically from a non-homogeneous Beatty sequence

## Source of Name

This entry was named for Samuel Beatty.