Zeros of the functions f(n) = Σi=0(−1)i(in−2i)

The main result of this paper is the following: the only zeros of the title function are at n = 3 and n = 12. This is achieved by means of the recursion function for f(n), viz. F(x) = x³ ? x ? 1 which has only one real root w. This turns out to be the fundamental unit of Q(w). From the norm equation of the units, N(w) = x³ + y³ + z³ ? 3xyz + 2x²z + xz² ? xy² ? yz² = 1, and the negative powers of w which are of binary form, the result follows. The paper concludes with two remarkable combinatorial identities.