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

%I A307232 #21 Jun 23 2019 17:00:40
%S A307232 1,1,3,73,6003,2318521,4132876803
%N A307232 a(n) is the number of n X n {0,1}-matrices (over the reals) that contain no zeros when squared.
%C A307232 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.
%H A307232 Wikipedia, <a href="https://en.wikipedia.org/wiki/Logical_matrix">Logical matrix</a>.
%e A307232 For n = 2, the a(2) = 3 solutions are
%e A307232   1 1    0 1    1 1
%e A307232   1 0    1 1    1 1
%t A307232 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;
%t A307232 Do[Print[a[n]], {n, 0, 5}] (* _Jean-François Alcover_, Jun 23 2019 *)
%o A307232 (MATLAB)
%o A307232 %Exhaustively searches all matrices
%o A307232 %from n = 1 to 5
%o A307232 result = zeros(1,5);
%o A307232 for n = 1:5
%o A307232 for m = 0:2^(n^2)-1
%o A307232     p = fliplr(dec2bin(m,n^2) - '0');
%o A307232     M = reshape(p,[n n]);
%o A307232     D = M^2;
%o A307232     if(isempty(find(D==0, 1)))
%o A307232         result(n) = result(n) + 1;
%o A307232     end
%o A307232 end
%o A307232 end
%Y A307232 Cf. A002720, A055601, A055602.
%Y A307232 A002416 is the total number of possible square binary matrices.
%Y A307232 A307248 gives a lower bound.
%K A307232 nonn,hard,more
%O A307232 0,3
%A A307232 _Christopher Cormier_, Mar 29 2019
%E A307232 a(6) from _Giovanni Resta_, May 29 2019