Here's a possibly-too-highbrow explanation to complement the nice simple one in the OP.<p>"As everyone knows", you get a Sierpinski triangle by taking the entries in Pascal's triangle mod 2. That is, taking <i>binomial coefficients</i> mod 2.<p>Now, here's a cute theorem about binomial coefficients and prime numbers: for any prime p, the number of powers of p dividing (n choose r) equals the number of <i>carries</i> when you write r and n-r in base p and add them up.<p>For instance, (16 choose 8) is a multiple of 9 but not of 27. 8 in base 3 is 22; when you add 22+22 in base 3, you have carries out of the units and threes digits.<p>OK. So, now, suppose you look at (x+y choose x) mod 2. This will be 1 exactly when <i>no</i> 2s divide it; i.e., when <i>no</i> carries occur when adding x and y in binary; i.e., when x and y never have 1-bits in the same place; i.e., when x AND y (bitwise) is zero.<p>And that's exactly what OP found!