the issue is that you're trying to use an odd-sized (3-pixel) pattern in a projection which is 2:1. Patterns, to keep the perfect lines, must have widths that are divisible by 2.
The reason it works for blackbeltdude is that he's got a 3-pixel wide window space with a 1 pixel wide gutter, making his pattern 4 pixels wide.
Similarly, over a wide enough area, even odd-width patterns will begin to become regular, however this will be every pair, because (odd width) + (odd width) = even width, which can repeat.
Basically, you can't treat the line as all the same. There are pixels which are mostly above the perfect line, and pixels which are below it. You have to treat these as though they are at different heights, because, well, they are.
here's some examples :
