A190482 Convex, obtuse, hexagonal lattice numbers.

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

Offset: 1

Author

Keenan Pepper, May 24 2011

Keywords

• nonn
• easy

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 nearest-neighbor lattice points in the set. (In other words, the natural polygon you get by following nearest-neighbor 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.)

Examples

			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.
		

Programs

PARI

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