Putnam 1974 B6

For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$ ), $\equiv 1$ (mod $3$ ), $\equiv 2$ (mod $3$ )? In other words, calculate $$s_{i,n}= \sum_{k\equiv i \;(\text{mod} 3)} \binom{n}{k}$$ for $i=0,1,2$ . Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$ . In particular, show the relationships among these three sums for $n = 1000$ .