Interleave sequence

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In mathematics, an interleave sequence is obtained by merging together two sequences.

Let S be a set, and let (xi) and (yi), 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 (zi),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.

[edit] Properties

  • The interleave sequence (zi) is convergent if and only if the sequences (xi) and (yi) 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).

This article incorporates material from Interleave sequence on PlanetMath, which is licensed under the GFDL.