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,&thinsp;1)&thinsp;×&thinsp;(0,&thinsp;1) to the interval (0,&thinsp;1). Different radixes give rise to different injections; the one for the binary numbers is called the Z-order curve or Morton code.