A227308 Given an equilateral triangular grid with side n consisting of n(n+1)/2 points, a(n) is the maximum number of points that can be painted so that, if any 3 of the painted ones are chosen, they do not form an equilateral triangle with sides parallel to the grid.

1, 2, 4, 6, 8, 12, 14, 18, 22, 26, 30, 34, 39, 44, 49

Offset: 1

Author

Heinrich Ludwig, Jul 06 2013

Keywords

• nonn
• hard
• more

Comments

Numbers found by an exhaustive computational search for all solutions. This sequence is complementary to A227116: A227116(n) + A227308(n) = n(n+1)/2.

Up to n=12 there is always a symmetric maximal solution. For n=13 and n=15 symmetric solutions contain at most a(n)-1 painted points. - Heinrich Ludwig, Oct 26 2013

Examples

			n = 11. At most a(11) = 30 points (X) of 66 can be painted, while 36 (.) must remain unpainted.
                .
               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 .
In this pattern there is no equilateral subtriangle with all vertices = X and sides parallel to the whole triangle.
		

Links

• Heinrich Ludwig, Illustration of a(2)..a(15)
• Giovanni Resta, Illustration of a(3)-a(14)

Crossrefs

Cf. A227116 (the complementary problem), A152125, A227133, A002717.

Programs

Mathematica

ivar[r_, c_] := r*(r-1)/2 + c; a[n_] := Block[{m, qq, nv = n*(n+1)/2, ne}, qq = Union[ Flatten[Table[{ivar[r, c], ivar[r-j, c], ivar[r, c+j]}, {r, 2, n}, {c, r - 1}, {j, Min[r - 1, r - c]}], 2], Flatten[Table[{ivar[r, c], ivar[r + j, c], ivar[r, c - j]}, {r, 2, n}, {c, 2, r}, {j, Min[c - 1, n - r]}], 2]]; ne = Length@qq; m = Table[0, {ne}, {nv}]; Do[m[[i, qq[[i]]]] = 1, {i, ne}]; Total@ Quiet@ LinearProgramming[ Table[-1, {nv}], m, Table[{2, -1}, {ne}], Table[{0, 1}, {nv}], Integers]]; Array[a, 9] (* Giovanni Resta, Sep 19 2013 *)

Extensions

a(12), a(13) from Heinrich Ludwig, Sep 02 2013

a(14) from Giovanni Resta, Sep 19 2013

a(15) from Heinrich Ludwig, Oct 26 2013