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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. 4
 1, 4, 6, 9, 16, 20, 26, 36, 42, 52, 64, 74, 86, 100, 114, 130 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS Manfred Scheucher, Python Script Peter J. Taylor, Java program to compute terms FORMULA 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 EXAMPLE 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 PROG (Java) 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 CROSSREFS Cf. A000769, A181019, A219760, A225623. Sequence in context: A010436 A085955 A266346 * A155569 A189484 A155567 Adjacent sequences:  A181015 A181016 A181017 * A181019 A181020 A181021 KEYWORD nonn,more,nice AUTHOR R. H. Hardin, Sep 30 2010 EXTENSIONS 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 STATUS approved

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Last modified July 9 11:06 EDT 2020. Contains 335543 sequences. (Running on oeis4.)