Growth function of the class of all circles in the plane

I know that the VC dimension for this problem is 3. My concern is about the growth function. The following bound is obtained using the VC dimension: $$m_{\mathcal{H}}(n)\le \sum_{k=0}^3\,{n\choose k}$$ When I simulate this hypothesis set up to $n= 10$, the maximum number of dichotomies I obtain is exactly $\sum_{k=0}^{3}\,{n\choose k}$. Is it by coincidence or is there a way to prove this?