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%I #55 Nov 21 2022 09:39:42
%S 1,2,4,6,8,10,12,14,17,20,22,25,28,31,34
%N Maximal number of points that can be placed on a triangular grid of side n so that no three of them are vertices of an equilateral triangle in any orientation.
%C Placing points on a triangular grid of side n, there are A000332(n + 3) triangles to be avoided.
%C The number k(n) of maximal solutions (reflections and rotations not counted) varies greatly: k(n) = 1, 1, 1, 1, 1, 3, 13, 129, 15, 2, 63, 3, 20, 1, ...
%C From _Elijah Beregovsky_, Nov 20 2022: (Start)
%C a(n) >= 3n-11.
%C This lower bound is given by the construction seen in the example section.
%C Conjecture: for n >= 11, a(n) = 3n-11. (End)
%e On a triangular grid of side 15, 34 points (X) can be placed so that no three of them form an equilateral triangle, regardless of its orientation.
%e X
%e . .
%e . X .
%e X . X .
%e . X . . X
%e X . . . X .
%e . X . . . . X
%e X . . . . . X .
%e . X . . . . . . X
%e X . . . . . . . X .
%e . X . . . . . . . . X
%e X . . . . . . . . . X .
%e . X . . . . . . . . . . X
%e . . . . . . . . . . . . X .
%e . . X X X X X X X X X X X . .
%Y Cf. A227308, A227116, A227133, A000332.
%K nonn,nice,hard,more
%O 1,2
%A _Heinrich Ludwig_, Apr 01 2014
%E a(14) from _Heinrich Ludwig_, Jun 20 2014
%E a(15) from _Heinrich Ludwig_, Jun 21 2016
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