How to calculate autocorrelation at certain lag?

I started learning time series analysis and I have one small "home task".
$$
X_t = Z_t + 0.5Z_{t-1} + 0.5Z_{t-2}, sigma^2=1 \
Find: gamma(0), gamma(2) text{ and autocorrelation function at lag 2, acf(2)}
$$
Where $gamma$ is autocovariance and $ Z_t sim iid(0,sigma^2) $.
I know that at lag 0:
$$
gamma(0) = sigma^2 sum_{i=0}^{q-k} beta_i^2
$$
and in the general form:
$$
gamma(k) = sigma^2 sum_{i=0}^{q-k}beta_ibeta_{i+k}
$$
I am computing $ gamma(0)$ and $gamma(2) $ as:
$$
gamma(0) = 1^2 + 0.5^2 + 0.5^2 = 1.5 \
gamma(2) = 1 * 0.5 = 0.5
$$
I know that autocorrelation function is:
$$
rho(2) = frac{gamma(2)}{gamma(0)}
$$
I think, that autocorrelation function at lag 2 must be then $ frac{0.5}{1.5} $. Am I doing something wrong? The automated test says that this is wrong. Can anyone please explain this to me?
[Edit]
The calculations were right. There was a problem with the automatic test system