fourier analysis - A sum involving sines

Consider a scaled sine function, $sin(2pi x/2^n)$, for some positive integer $n$. For this, I have the following linear combination.

$$ sum_{x=1}^{2^{n-2}} c_x sin(2pi x/2^n).$$
(The upper limit to the sum is $2^{n-2}$.)

The question is whether there exist $c_x in {0, pm 1, pm 2}$, not all $0$, that make the above expression $0$, for infinitely many $n$?

If it helps, the above came up in a computation concerning the discrete Fourier Transform.