Let $\alpha$ be an irrational real number; the behavior of the sum $S_N(\alpha):=(-1)^{[\alpha]}+(-1)^{[2\alpha]}+\cdots+(-1)^{[N\alpha]}$ depends on the continued fraction expansion of $\alpha/2$. Since the continued fraction expansion of $\sqrt{2}/2$ has bounded partial quotients, $S_N(\sqrt{2})=O(\log(N))$ and this bound is best possible. The partial quotients of the continued fraction expansion of $e$ grow slowly and thus $S_N(2e)=O(\frac{\log(N)^2}{\log\log(N)^2})$, again best possible. The partial quotients of the continued fraction expansion of $e/2$ behave similarly as those of $e$. Surprisingly enough $S_N(e)=O(\frac{\log(N)}{\log\log(N)})$.