Interleave sequence

Short description: Result of merging two sequences by perfect shuffling

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

Let [math]\displaystyle{ S }[/math] be a set, and let [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_i) }[/math], [math]\displaystyle{ i=0,1,2,\ldots, }[/math] be two sequences in [math]\displaystyle{ S. }[/math] The interleave sequence is defined to be the sequence [math]\displaystyle{ x_0, y_0, x_1, y_1, \dots }[/math]. Formally, it is the sequence [math]\displaystyle{ (z_i), i=0,1,2,\ldots }[/math] given by

[math]\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} }[/math]

Properties

• The interleave sequence [math]\displaystyle{ (z_i) }[/math] is convergent if and only if the sequences [math]\displaystyle{ (x_i) }[/math] and [math]\displaystyle{ (y_i) }[/math] 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.