

A307232


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


1




OFFSET

0,3


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.


LINKS



EXAMPLE

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


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;


PROG

(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


CROSSREFS

A002416 is the total number of possible square binary matrices.


KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



