Interleave sequence

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

Let $S$ be a set, and let $(x_{i})$ and $(y_{i})$, $i=0,1,2,\ldots ,$ 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 $(z_{i}),i=0,1,2,\ldots$ given by

$z_{i}:={\begin{cases}x_{i/2}&{\text{ if }}i{\text{ is even,}}\\y_{(i+1)/2}&{\text{ if }}i{\text{ is odd.}}\end{cases}}$

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.