Interleave sequence

In mathematics, an interleave sequence is obtained by merging or shuffling 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.