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We first derive the function from <math>(x,y) \in \mathbb{Z}^2</math> to <math>z \in \mathbb{Z}</math> as shown in Figure 2. | We first derive the function from <math>(x,y) \in \mathbb{Z}^2</math> to <math>z \in \mathbb{Z}</math> as shown in Figure 2. | ||
Let <math>(x,y)</math> be the coordinates in <math>\{(0,0), (1, 0), (1,1), ...\}</math> which will map to <math>\ | Let <math>(x,y)</math> be the coordinates in <math>\{(0,0), (1, 0), (1,1), ...\}</math> which will map to <math>\{0, 1, 2, ...\}</math>. | ||
First note that <math>\sum_{0}^{k}i = \frac{(k)(k+1)}{2}</math>. | First note that <math>\sum_{0}^{k}i = \frac{(k)(k+1)}{2}</math>. | ||
Thus the number of elements in columns <math>0, ..., x-1</math> is <math>(x)(x+1)/2</math>. | Thus the number of elements in columns <math>0, ..., x-1</math> is <math>(x)(x+1)/2</math>. | ||