Interleave sequence

In mathematics, an interleave sequence is obtained by merging two sequences.

Let $S$ be a set, and let $(x_{i})$ and $(y_{i})$, $i=0,1,2,...,$ be two sequences in $S.$ The interleave sequence is defined to be the sequence $x_{0},y_{0},x_{1},y_{1},\dots .$ Formally, it is the sequence $(z_{i}),i=0,1,2,...$ given by

$z_{i}:=\left\{{\begin{matrix}x_{k}&{\mbox{ if }}i=2k{\mbox{ is even,}}\\y_{k}&{\mbox{ if }}i=2k+1{\mbox{ is odd.}}\end{matrix}}\right.$

Properties

• The interleave sequence $(z_{i})$ is convergent if and only if the sequences $(x_{i})$ and $(y_{i})$ are convergent and have the same limit.

• Consider two real numbers a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b, which will determine a third number c, also greater than zero and smaller than 1. In this way one obtains an injection from the square (0, 1)×(0, 1) to the interval (0, 1). Different radixes give rise to different injections; the one for the binary numbers is called the Z-order curve or Morton code.

References

This article incorporates material from Interleave sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.