A225623 -id:A225623 - cp's OEIS frontend

A181018 Maximum number of 1's in an n X n binary matrix with no three 1's adjacent in a line along a row, column or diagonally.

1, 4, 6, 9, 16, 20, 26, 36, 42, 52, 64, 74, 86, 100, 114, 130

Offset: 1

R. H. Hardin, Sep 30 2010

Diagonal of A181019.
Three or more "1"s may be adjacent in an L-shape or step shape (cf. bottom of first example) or 2 X 2 square (top right of 2nd example) or similar. One possible (not always optimal) solution is therefore to fill the square with 2 X 2 squares of "1"s, separated by rows of "0"s: this yields the lower bound (n - floor(n/3))^2 = ceiling(2n/3)^2 given in FORMULA. I conjecture that this is optimal for n = 2 (mod 3) and that a(n) ~ (2n/3)^2. For n = 3k, the array can be filled with 2k(2k+1) "1"s by repeating the optimal solution for n = 3 on the diagonal, and filling the rest with 2 X 2 blocks separated by rows of "0"s, cf. the 4th example for 6 X 6. - M. F. Hasler, Jul 17 2015 [Conjecture proved to be wrong, see below. - M. F. Hasler, Jan 19 2016]
74 <= a(12) <= 77. - Manfred Scheucher, Jul 23 2015
You can repeat a 4 X 2 block [1100; 0011] infinitely in both directions and then crop the needed square. That gives ceiling(n^2/2). It eventually surpasses the solutions we've found so far: at 17*17 the pattern above gives 12*12=144 but this one ceiling(17*17/2)=145. The credit for finding this goes to Jaakko Himberg. - Juhani Heino, Aug 11 2015

			Some solutions for 6 X 6:
  0 1 1 0 1 1    0 1 1 0 1 1    0 1 1 0 1 1    0 1 1 0 1 1
  1 0 1 0 0 1    1 0 1 0 1 1    1 0 1 0 0 1    1 0 1 0 1 1
  1 1 0 0 1 0    1 1 0 0 0 0    1 1 0 0 1 0    1 1 0 0 0 0
  0 0 0 0 1 1    0 0 0 0 1 1    0 0 0 0 1 1    0 0 0 0 1 1
  1 0 1 1 0 1    1 0 1 1 0 1    1 1 0 1 0 1    1 1 0 1 0 1
  1 1 0 1 1 0    1 1 0 1 1 0    1 1 0 1 1 0    1 1 0 1 1 0
A solution with 73 ones for 12 X 12 (I replaced "0" with "." for readability):
  1 1 . 1 1 . 1 1 . 1 . 1
  1 1 . . 1 1 . 1 1 . 1 1
  . . . 1 . . . . . . 1 .
  1 1 . 1 . 1 . 1 1 . . 1
  . 1 1 . . 1 1 . . 1 1 .
  1 . . . 1 . 1 . 1 . . 1
  1 1 . . 1 1 . . 1 . 1 .
  . 1 . 1 . 1 . 1 . . 1 1
  1 . . 1 1 . . 1 1 . . 1
  . 1 . . . . 1 . 1 . 1 .
  1 1 . 1 1 . 1 1 . . 1 1
  1 . 1 . 1 1 . 1 . 1 . 1
- _Manfred Scheucher_, Jul 23 2015
An optimal solution with 74 ones (denoted by O) for 12 X 12 (also symmetric):
  O . O . O . O O . O O .
  O O . O O . . . O O . O
  . O . O . O O . . . O O
  O . . . O O . O O . O .
  . O O . . . O . . . . O
  O O . O O . O . O O . .
  . . O O . O . O O . O O
  O . . . . O . . . O O .
  . O . O O . O O . . . O
  O O . . . O O . O . O .
  O . O O . . . O O . O O
  . O O . O O . O . O . O - _Giovanni Resta_, Jul 29 2015

Manfred Scheucher, Python Script
Peter J. Taylor, Java program to compute terms

Cf. A000769, A181019, A219760, A225623.

See Taylor link
(MATLAB with CPLEX)
function v = A181018(n)
%
Grid = [1:n]' * ones(1,n) + n*ones(n,1)*[0:n-1];
f = -ones(n^2,1);
A = sparse(4*(n-2)*(n-1),n^2);
count = 0;
for i =1:n
  for j = 1:n-2
    count = count+1;
    A(count, [Grid(i,j),Grid(i,j+1),Grid(i,j+2)]) = 1;
  end
end
for i = 1:n-2
  for j = 1:n
    count = count+1;
    A(count, [Grid(i,j),Grid(i+1,j),Grid(i+2,j)]) = 1;
  end
end
for i = 1:n-2
  for j = 1:n-2
    count = count+2;
    A(count-1,[Grid(i,j+2),Grid(i+1,j+1),Grid(i+2,j)]) = 1;
    A(count, [Grid(i,j),Grid(i+1,j+1),Grid(i+2,j+2)]) = 1;
  end
end
b = 2*ones(4*(n-2)*(n-1),1);
[x,v,exitflag,output] = cplexbilp(f,A,b);
end;
for n = 1:11
  A(n) = A181018(n);
end
A % Robert Israel, Jan 14 2016

a(n) >= ceiling(2n/3)^2; a(3k) >= A002943(k) = 2k(2k+1). - M. F. Hasler, Jul 17 2015; revised by Juhani Heino, Aug 11 2015

a(n) >= ceiling(n^2/2). - Juhani Heino, Aug 11 2015

a(11)-a(12) from M. F. Hasler, Jul 20 2015
a(12) deleted by Manfred Scheucher, Jul 23 2015
a(12) from Giovanni Resta, Jul 29 2015
PARI code (which implemented a conjectured formula shown to underestimate) deleted by Peter J. Taylor, Jan 06 2016
a(13)-a(15) from Peter J. Taylor, Jan 09 2016
a(16) from Peter J. Taylor, Jan 14 2016

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A181018 Maximum number of 1's in an n X n binary matrix with no three 1's adjacent in a line along a row, column or diagonally.

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