Partial sums, you mean?
. Suppose n is 0 or 3 mod 4. Then n(n+1)/2 is even. Otherwise, n(n+1)/2 is odd. This should tell you how to handle the first sum. Now suppose n is 0 mod 4. Then is 1. If n is 2 mod 4, then it is -1. Otherwise, it is 0. This will let you handle the second sum.