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A190482 Convex, obtuse, hexagonal lattice numbers.

%I A190482 #17 Jul 22 2025 12:03:41
%S A190482 7,10,12,13,14,16,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,
%T A190482 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,
%U A190482 58,59,60,61
%N A190482 Convex, obtuse, hexagonal lattice numbers.
%C A190482 These are the integers N for which it is possible to arrange N points in a subset of a hexagonal lattice such that:
%C A190482 1. The points are not all collinear.
%C A190482 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.)
%C A190482 3. The convex hull contains only obtuse angles, which will necessarily be 120 degrees.
%C A190482 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".
%C A190482 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.)
%e A190482 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.
%e A190482 It's unclear whether 1 should be included, but the strict definition given above excludes N = 1.
%o A190482 (PARI) a(n)=if(n>6,n+11,[7,10,12,13,14,16][n]) \\ _Charles R Greathouse IV_, Aug 26 2011
%K A190482 nonn,easy
%O A190482 1,1
%A A190482 _Keenan Pepper_, May 24 2011