

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

Table of n, a(n) for n=0..6.
Wikipedia, Logical matrix.


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;
Do[Print[a[n]], {n, 0, 5}] (* JeanFrançois Alcover, Jun 23 2019 *)


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

Cf. A002720, A055601, A055602.
A002416 is the total number of possible square binary matrices.
A307248 gives a lower bound.
Sequence in context: A012810 A020517 A119017 * A002667 A145675 A336873
Adjacent sequences: A307229 A307230 A307231 * A307233 A307234 A307235


KEYWORD

nonn,hard,more


AUTHOR

Christopher Cormier, Mar 29 2019


EXTENSIONS

a(6) from Giovanni Resta, May 29 2019


STATUS

approved



