A307248 -id:A307248 - cp's OEIS frontend

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

Christopher Cormier, Mar 29 2019

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.

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

Wikipedia, Logical matrix.

Cf. A002720, A055601, A055602.
A002416 is the total number of possible square binary matrices.
A307248 gives a lower bound.

%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

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 *)

a(6) from Giovanni Resta, May 29 2019

cp's OEIS Frontend

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

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