

A190482


Convex, obtuse, hexagonal lattice numbers


14



7, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61
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OFFSET

1,1


COMMENTS

These are the integers N for which it is possible to arrange N points in a subset of a hexagonal lattice such that:
1. The points are not all collinear.
2. The convex hull of the points consists entirely of line segments between nearestneighbor lattice points in the set. (In other words, the natural polygon you get by following nearestneighbor connections around the outside of the set is a convex polygon.)
3. The convex hull contains only obtuse angles, which will necessarily be 120 degrees.
These can also be thought of as "convenient bundle of cylinders" numbers, because if you are carrying a bundle of N parallel, identical cylinders, only if N is in this list can they be arranged in such a way that there are no "gaps", and also no single cylinders sticking out of the pattern creating "bulges".
The greatest integer that is absent from this list is 17. (This can be proved by exhaustion  it's possible to find infinite sequences of patterns that eventually cover every possible remainder of N mod 6, and 17 is the last N that cannot be achieved.)


LINKS

Table of n, a(n) for n=1..50.


EXAMPLE

For N = 2, 3, 4, 5, or 6, it is impossible to arrange N cylinders in a neat bundle with no bulges or dents, so they are not in the list. But for N = 7, six cylinders can surround a central one, so 7 is in the list.
It's unclear whether 1 should be included, but the strict definition given above excludes N = 1.


PROG

(PARI) a(n)=if(n>6, n+11, [7, 10, 12, 13, 14, 16][n]) \\ Charles R Greathouse IV, Aug 26 2011


CROSSREFS

Sequence in context: A247868 A048792 A024585 * A128783 A024586 A050419
Adjacent sequences: A190479 A190480 A190481 * A190483 A190484 A190485


KEYWORD

nonn,easy


AUTHOR

Keenan Pepper, May 24 2011


STATUS

approved



