Minimal Edge Series

Imagine having a collection of unit edges and being asked to construct as many regular hexagonal enclosures as possible while using as few of the (presumably expensive) edges as you are able. Clearly, the problem is trivial unless one allows sharing of edges in which case it turns out that an optimal strategy is to simply start tiling the plane in a spiral manner until you reach the desired number of enclosures. This leads to the spiral hexagonal series (counting cumulative edges used) of 6,11,15,19,23,27,30,... The resulting pattern of additional edges is shown in the figure above where the yellow and green regions continue in the obvious way off to infinity.

See Redheads Matchbox Puzzles number 8 for the inspiration to this problem, Spiral Polygon Series for early musing on related series and Minimising Polygonal Edges for more recent generalisations and proofs.