Interleave sequence

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

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

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

Properties

• The interleave sequence ${\displaystyle (z_{i})}$ is convergent if and only if the sequences ${\displaystyle (x_{i})}$ and ${\displaystyle (y_{i})}$ are convergent and have the same limit.^[1]
• 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.^[2]