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 $\mathcal B_x$ 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


Also see


Source of Name

This entry was named for Samuel Beatty.


Sources