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A227133

Given a square grid with side n consisting of n^2 cells (or points), a(n) is the maximum number of points that can be painted so that no four of the painted ones form a square with sides parallel to the grid.

1, 3, 7, 12, 17, 24, 32, 41, 51, 61, 73, 85, 98

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OFFSET

1,2

COMMENTS

a(1) through a(9) were found by an exhaustive computational search for all solutions. This sequence is complementary to A152125: A152125(n) + A227133(n) = n^2.

A064194(n) is a lower bound on a(n) (see the comments in A047999). - N. J. A. Sloane, Jan 18 2016

a(11) >= 71 (by extending the n=10 solution towards the southeast). - N. J. A. Sloane, Feb 12 2016

a(11) >= 73, a(12) >= 85, a(13) >= 98, a(14) >= 112, a(15) >= 127, a(16) >= 142 (see links). These lower bounds were obtained using tabu search and simulated annealing via the Ascension Optimization Framework. - Peter Karpov, Feb 22 2016; corrected Jun 04 2016

Note that n is the number of cells along each edge of the grid. The case n=1 corresponds to a single square cell, n=2 to a 2 X 2 array of four square cells. The standard chessboard is the case n=8. It is easy to get confused and to think of the case n=2 as a 3 X 3 grid of dots (the vertices of the squares in the grid). Don't think like that! - N. J. A. Sloane, Apr 03 2016

a(12) = 85 and a(13) = 98 were obtained with a MIP model, solved with Gurobi in 141 days on 32 cores. - Simon Felix, Nov 22 2019

LINKS

Table of n, a(n) for n=1..13.

Peter Karpov, InvMem, Item 20 [Link added by N. J. A. Sloane, Feb 24 2016]

Peter Karpov, Ascension Optimization Framework [Link added by N. J. A. Sloane, Feb 24 2016]

Peter Karpov, Best configurations known for n = 11..16

Giovanni Resta, Illustration of a(2)-a(10)

Giovanni Resta, Individual illustration for a(8)

EXAMPLE

n=9. A maximum of a(9) = 51 points (X) of 81 can be painted while 30 (.) must be left unpainted. The following 9 X 9 square is an example:

. 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 X . . X X . X

X X . X . X . X X

. X X X X X X X .

Here there is no subsquare with all vertices = X and having sides parallel to the axes.

MATHEMATICA

a[n_] := Block[{m, qq, nv = n^2, ne}, qq = Flatten[1 + Table[n*x + y + {0, s, s*n, s*(n + 1)}, {x, 0, n-2}, {y, 0, n-2}, {s, Min[n-x, n-y] -1}], 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[{3, -1}, {ne}], Table[{0, 1}, {nv}], Integers]]; Array[a, 8] (* Giovanni Resta, Jul 14 2013 *)

CROSSREFS

Cf. A152125 (the complementary problem), A000330, A240443 (when all squares must be avoided, not just those aligned with the grid).