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%I #76 Jul 08 2018 19:59:04
%S 1,4,6,9,16,20,26,36,42,52,64,74,86,100,114,130
%N 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.
%C Diagonal of A181019.
%C 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]
%C 74 <= a(12) <= 77. - _Manfred Scheucher_, Jul 23 2015
%C 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
%H Manfred Scheucher, <a href="/A181018/a181018.py.txt">Python Script</a>
%H Peter J. Taylor, <a href="/A181018/a181018.java.txt">Java program to compute terms</a>
%F 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
%F a(n) >= ceiling(n^2/2). - _Juhani Heino_, Aug 11 2015
%e Some solutions for 6 X 6:
%e 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1
%e 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1
%e 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0
%e 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1
%e 1 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1
%e 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0
%e A solution with 73 ones for 12 X 12 (I replaced "0" with "." for readability):
%e 1 1 . 1 1 . 1 1 . 1 . 1
%e 1 1 . . 1 1 . 1 1 . 1 1
%e . . . 1 . . . . . . 1 .
%e 1 1 . 1 . 1 . 1 1 . . 1
%e . 1 1 . . 1 1 . . 1 1 .
%e 1 . . . 1 . 1 . 1 . . 1
%e 1 1 . . 1 1 . . 1 . 1 .
%e . 1 . 1 . 1 . 1 . . 1 1
%e 1 . . 1 1 . . 1 1 . . 1
%e . 1 . . . . 1 . 1 . 1 .
%e 1 1 . 1 1 . 1 1 . . 1 1
%e 1 . 1 . 1 1 . 1 . 1 . 1
%e - _Manfred Scheucher_, Jul 23 2015
%e An optimal solution with 74 ones (denoted by O) for 12 X 12 (also symmetric):
%e O . O . O . O O . O O .
%e O O . O O . . . O O . O
%e . O . O . O O . . . O O
%e O . . . O O . O O . O .
%e . O O . . . O . . . . O
%e O O . O O . O . O O . .
%e . . O O . O . O O . O O
%e O . . . . O . . . O O .
%e . O . O O . O O . . . O
%e O O . . . O O . O . O .
%e O . O O . . . O O . O O
%e . O O . O O . O . O . O - _Giovanni Resta_, Jul 29 2015
%o (Java) See Taylor link
%o (MATLAB with CPLEX)
%o function v = A181018(n)
%o %
%o Grid = [1:n]' * ones(1,n) + n*ones(n,1)*[0:n-1];
%o f = -ones(n^2,1);
%o A = sparse(4*(n-2)*(n-1),n^2);
%o count = 0;
%o for i =1:n
%o for j = 1:n-2
%o count = count+1;
%o A(count, [Grid(i,j),Grid(i,j+1),Grid(i,j+2)]) = 1;
%o end
%o end
%o for i = 1:n-2
%o for j = 1:n
%o count = count+1;
%o A(count, [Grid(i,j),Grid(i+1,j),Grid(i+2,j)]) = 1;
%o end
%o end
%o for i = 1:n-2
%o for j = 1:n-2
%o count = count+2;
%o A(count-1,[Grid(i,j+2),Grid(i+1,j+1),Grid(i+2,j)]) = 1;
%o A(count, [Grid(i,j),Grid(i+1,j+1),Grid(i+2,j+2)]) = 1;
%o end
%o end
%o b = 2*ones(4*(n-2)*(n-1),1);
%o [x,v,exitflag,output] = cplexbilp(f,A,b);
%o end;
%o for n = 1:11
%o A(n) = A181018(n);
%o end
%o A % _Robert Israel_, Jan 14 2016
%Y Cf. A000769, A181019, A219760, A225623.
%K nonn,more,nice
%O 1,2
%A _R. H. Hardin_, Sep 30 2010
%E a(11)-a(12) from _M. F. Hasler_, Jul 20 2015
%E a(12) deleted by _Manfred Scheucher_, Jul 23 2015
%E a(12) from _Giovanni Resta_, Jul 29 2015
%E PARI code (which implemented a conjectured formula shown to underestimate) deleted by _Peter J. Taylor_, Jan 06 2016
%E a(13)-a(15) from _Peter J. Taylor_, Jan 09 2016
%E a(16) from _Peter J. Taylor_, Jan 14 2016
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