A307232 a(n) is the number of n X n {0,1}-matrices (over the reals) that contain no zeros when squared.

1, 1, 3, 73, 6003, 2318521, 4132876803

Offset: 0

Author

Christopher Cormier, Mar 29 2019

Keywords

• nonn
• hard
• more

Comments

For every n, there are trivial solutions where an entire row is filled with 1's and an entire column is filled with 1's, and the column index is equal to the row index. This easily follows from the nature of matrix multiplication. Every matrix that has at least one of these row/column pairs along with any other 1's is also a solution because there are no negative numbers involved here. The number of trivial solutions is given by A307248.

Examples

			For n = 2, the a(2) = 3 solutions are
  1 1    0 1    1 1
  1 0    1 1    1 1
		

Links

• Wikipedia, Logical matrix.

Crossrefs

Cf. A002720, A055601, A055602.

A002416 is the total number of possible square binary matrices.

A307248 gives a lower bound.

Programs

MATLAB

%Exhaustively searches all matrices
%from n = 1 to 5
result = zeros(1,5);
for n = 1:5
for m = 0:2^(n^2)-1
    p = fliplr(dec2bin(m,n^2) - '0');
    M = reshape(p,[n n]);
    D = M^2;
    if(isempty(find(D==0, 1)))
        result(n) = result(n) + 1;
    end
end
end

Mathematica

a[n_] := Module[{b, iter, cnt = 0}, iter = Sequence @@ Table[{b[k], 0, 1}, {k, 1, n^2}]; Do[If[FreeQ[MatrixPower[Partition[Array[b, n^2], n], 2], 0], cnt++], iter // Evaluate]; cnt]; a[0] = 1;
Do[Print[a[n]], {n, 0, 5}] (* Jean-François Alcover, Jun 23 2019 *)

Extensions

a(6) from Giovanni Resta, May 29 2019