Here is a pattern to list the rationals between 0 and 1:
0, 1, 1/2, 1/3, 2/3, 1/4, 2/4, 3/4,...
I find it simpler than the usual method based on the diagonals of the square lattice, but I guess it appears less frequently because it only lists the rationals between 0 and 1, rather than all...