Saturday, March 19, 2016

Hex Grid Area

Consider the hex grid. Take a single hex on that grid; it has perimeter 0, and area 1. Then consider the ring of hexes around that hex; it has perimeter 6 and area 7. As you consider larger and larger hexagonal areas, an interesting pattern emerges:

size          1: perimeter=0,                   area=1
size         10: perimeter=54,                 area=271
size        100: perimeter=594,               area=29701
size       1000: perimeter=5994,             area=2997001
size      10000: perimeter=59994,           area=299970001
size     100000: perimeter=599994,         area=29999700001
size    1000000: perimeter=5999994,       area=2999997000001
size   10000000: perimeter=59999994,     area=299999970000001
size  100000000: perimeter=599999994,   area=29999999700000001
size 1000000000: perimeter=5999999994, area=2999999997000000001

I've just learned that the complete sequence of hexagonal areas (not just the powers of ten) is registered in the OEIS as A003215 (aka the "crystal ball sequence for hexagonal lattice").

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