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.

1, 2, 4, 6, 8, 10, 12, 14, 17, 20, 22, 25, 28, 31, 34

Offset: 1

Author

Heinrich Ludwig, Apr 01 2014

Keywords

• nonn
• nice
• hard
• more

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)

Examples

			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

Cf. A227308, A227116, A227133, A000332.

Extensions

a(14) from Heinrich Ludwig, Jun 20 2014

a(15) from Heinrich Ludwig, Jun 21 2016