# Definition:Beatty Sequence

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## 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 $\BB_x$ be the **Beatty sequence** on $x$.

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

## Also known as

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

## Also see

## Source of Name

This entry was named for Samuel Beatty.

## Sources

- Weisstein, Eric W. "Beatty Sequence." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/BeattySequence.html