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 $(y_i)$, $i=0,1,2,\dots ,$ 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 $(z_i),i=0,1,2,\dots$ given by

$z_i:=\{\begin{array}{cc}{x}_{i/2}& \text{ if }i\text{ is even,}\\ y_{(i-1)/2}& \text{ if }i\text{ is odd.}\end{array}$

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.