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

[Anzewill]

Anyone can give me a example of a bijective fuction from N to N^2?

 

[HallsofIvy]

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.

 

[Martin Rattigan]

Or for a simple formula try $(m,n)\rightarrow 2^m(2n+1)-1$.