ISL 2011 N4

For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences $$t(n+a)-t(n), t(n+a+1)-t(n+1), \ldots, t(n+2 a-1)-t(n+a-1)$$ are divisible by 4 .