Added "graph-theory" tag (see my answer below); hope that's OK.
Maximal number of edges and triangular cells for n points in a triangular lattice
Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This sequence appears to start 0, 1, 3, 5, 7, 9, 12, 14, 16...
What is the maximum number of triangular lattice cells? (Not the number of all triangles, just the number of smallest possible equilateral triangles in the lattice.) This sequence appears to start 0, 0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 13...
http://oeis.org/A047932 is related to the first sequence but I have no proof it's the same. (There might be some other way of arranging the pennies that yields a higher number of contacts. A047932 is a lower bound on my sequence.) I can't find any OEIS sequences relevant to the second one.