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