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

1,2

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)

Links

Table of n, a(n) for n=1..15.

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

Cf. A227308, A227116, A227133, A000332.

Sequence in context: A063459 A186329 A062417 * A076828 A276106 A321501

Adjacent sequences: A240111 A240112 A240113 * A240115 A240116 A240117

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