If you have ever tried making patterns with a collection of coins, you have probably noticed that you can make hexagons in a natural way by packing circles as tightly as possible. The figure below shows how 19 circles fit into a hexagonal shape with 3 circles on each edge. LetH(n)

be the number of circles you need to form a hexagon withncircles on each edge. From the figure below, it is clear thatH(2) = 7

andH(3) = 19.

It can be shown that increasing the number of circles on each edge gives the following recurrence relation:

H(n) =1ifn= 1H(n?1) + 6n?6ifn> 1.

CalculateH(5).

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