Putnam 1992 A5

For each positive integer $n$, let $a_n = 0$ (or 1) if the number of 1's in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that $$a_{k+j} = a_{k+m+j} = a_{k+2m+j},$$ for $0 \leq j \leq m-1$.