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A240114
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.
5
1, 2, 4, 6, 8, 10, 12, 14, 17, 20, 22, 25, 28, 31, 34
OFFSET
1,2
COMMENTS
Placing points on a triangular grid of side n, there are A000332(n + 3) triangles to be avoided.
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, ...
From Elijah Beregovsky, Nov 20 2022: (Start)
a(n) >= 3n-11.
This lower bound is given by the construction seen in the example section.
Conjecture: for n >= 11, a(n) = 3n-11. (End)
EXAMPLE
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.
X
. .
. X .
X . X .
. X . . X
X . . . X .
. X . . . . X
X . . . . . X .
. X . . . . . . X
X . . . . . . . X .
. X . . . . . . . . X
X . . . . . . . . . X .
. X . . . . . . . . . . X
. . . . . . . . . . . . X .
. . X X X X X X X X X X X . .
CROSSREFS
KEYWORD
nonn,nice,hard,more
AUTHOR
Heinrich Ludwig, Apr 01 2014
EXTENSIONS
a(14) from Heinrich Ludwig, Jun 20 2014
a(15) from Heinrich Ludwig, Jun 21 2016
STATUS
approved