Given an ordered sequence of points P = {p1, p2, ..., pn}, we consider all contiguous subsequences Pi,j := {pi, ..., pj} of P and the set T of distinct Delaunay triangles within their Delaunay triangulations. For arbitrary point sets and orderings, we give an O(n^2) bound on |T|. Furthermore, for arbitrary point sets in uniformly random order, we give two proofs of a T(n log n) bound on E[|T|].