

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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, ...
a(n) >= 3n11.
This lower bound is given by the construction seen in the example section.
Conjecture: for n >= 11, a(n) = 3n11. (End)


LINKS



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



EXTENSIONS



STATUS

approved



