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}

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 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. ==References==

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