Bijective Function from N to N^2: Examples and Help

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A bijective function from N to N^2 can be constructed by arranging natural numbers in a grid, where the cell at position (m,n) corresponds to the pair (m,n). The zigzag pattern starts at (1,1) and moves horizontally, then diagonally, covering all pairs systematically. An alternative formula for this function is (m,n) → 2^m(2n+1)-1. This approach ensures that every natural number is uniquely paired with a coordinate in N^2. The discussion highlights both a visual method and a mathematical formula for establishing the bijection.
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Anyone can give me a example of a bijective fuction from N to N^2?
 
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I can't give you a simple "formula" but here is how to get a bijective function:

Write the numbers 1, 2, 3, ... horizontally and to left of "1" and slightly below write 1, 2, 3, ... vertically so that the cell below "m" and to the right of "n" is the pair (m,n). Now start at (1, 1) and "zigzag" through that array. That is, go from (1,1) horizontally to (2, 1) then diagonally, down and left, to (1,2), down to (1, 3), diagonally up and right to (2,3) and (3, 1), right to (4, 1), diagonally down to (3,2), etc.
 
Or for a simple formula try (m,n)\rightarrow 2^m(2n+1)-1.
 

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