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 $\BB_x$ 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

• Beatty's Theorem

• Definition:Non-Homogeneous Beatty Sequence

• Definition:Lower Wythoff Sequence
• Definition:Upper Wythoff Sequence

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