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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.
8
1, 2, 4, 6, 8, 12, 14, 18, 22, 26, 30, 34, 39, 44, 49
OFFSET
1,2
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
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A227116 (the complementary problem), A152125, A227133, A002717.
Sequence in context: A324102 A089623 A089681 * A214294 A233578 A057220
KEYWORD
nonn,hard,more
AUTHOR
Heinrich Ludwig, Jul 06 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
STATUS
approved