[math-fun] Puzzle: Can you ID this sequence?

Here are the first 20 terms of an infinite sequence of nonnegative integers: 0,1,0,0,1,0,2,0,1,1,2,2,1,0,0,1,0,1,0,2,... Can you ID this (somewhat novel) sequence? (Hint: It's based on pi/4, and each term is the floor of a certain finite expression in closed form.) --Dan