Interleave sequence

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

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

z i := { x i / 2  if  i  is even, y ( i − 1 ) / 2  if  i  is odd. {\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}}}

• The interleave sequence ( z i ) {\displaystyle (z_{i})} is convergent if and only if the sequences ( x i ) {\displaystyle (x_{i})} and ( y i ) {\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]

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