Some patterns here look intriguing. Like
f(2^n) = 2^(2^(n-1))
for one, and
f(6n) = 10^n
for another. Do these extend?
I'm doubtful about the second. But the first one does. Define Fn := the field extension of Q generated by a primitive n'th root of unity. (This is sometimes called the n'th cyclotomic field). And define wn := such a primitive root. Oh, and define Pn := the standard regular n-gon. Now consider any subset A of P(2n) that has sum 0. Partition its vertices into A0, the ones that also appear in Pn, and A1, the ones that also appear in wn*Pn. And let a0 := sum(A0) and a1 := sum(A1). On the one hand, we have a0+a1 = 0. On the other, a0 is an element of Fn and a1 is w(2n) times an element of Fn. Since w(2n) isn't in Fn, this is only possible if both elements are 0. That is, a0=a1=0. Hence A0 and A1 are both 0-sum subsets of P(n), the latter rotated a bit. The number of ways to do this is clearly the square of the number of 0-sum subsets of P(n). Hence our sequence obeys the relation u(2n) = u(n)^2. It's not hard to calculate u(2) :-), so we're done. Let's generalize. Let p be prime. Then consider a subset A of P(pn) with sum 0. Partition its vertices into A0, ..., A(p-1) and let a0, ..., a(p-1) be the sums of the subsets. Then we get a polynomial in w(pn), with coefficients in F(n), having degree at most p-1, equal to 0. Provided the degree of w(pn) over F(n) is p -- which is true sometimes but not always -- we can proceed as before. What is the degree of w(pn) over F(n)? It's the same as the degree of the field extension F(pn):F(n), and that in turn is phi(pn)/phi(n), which is p-1 if p doesn't divide n and p if p does divide n. So: Theorem: If p divides n, then u(pn) = u(n)^p. We'll be done if we can find u(n) when n is a product of distinct primes. I don't see how to do that right now, but I suspect it's not all that hard. (Famous last words.) -- g