

A121231


Number of n X n binary matrices M (that is, real matrices with entries 0 and 1) such that M^2 is also a binary matrix.


6




OFFSET

0,2


COMMENTS

Comments from Brendan McKay, Aug 21 2006: Equivalently, directed graphs (simple but loops allowed) without a few small forbidden subgraphs (those allowing 2 distinct paths of length 2 from vertex x to vertex y for some x,y; I think there are 6 possibilities). One can also consider isomorphism classes of those digraphs.
Comment from Rob Pratt, Aug 03 2008: A121294 provides a lower bound on the maximum number of 1's in such a matrix M. There are cases where a higher number is reached; the following 5 X 5 matrix has 11 ones and its square is binary:
0 0 1 0 0
0 0 0 0 1
1 1 0 0 1
1 1 0 1 0
1 1 0 1 0.
The optimal values seem to match A070214, verified for n <= 7.
Term (5,1) of the nth power of the 5 X 5 matrix shown is A001045(n), the Jacobsthal sequence.  Gary W. Adamson, Oct 03 2008
a(n) >= A226321(n).


LINKS

Table of n, a(n) for n=0..7.
Eric Weisstein's World of Mathematics, Background information about adjacency matrices
E. W. Weisstein, (0,1)Matrix, MathWorld. [P. Petsie, Aug 03 2008]
Wikipedia, Background information about adjacency matrices
Index entries for matrices, binary, which are squares


CROSSREFS

Cf. A226321, A225371, A055084, A052264, A051589, A069452, A053304, A001045, A121294, A070214.
Sequence in context: A051255 A120445 A003088 * A122527 A039747 A049531
Adjacent sequences: A121228 A121229 A121230 * A121232 A121233 A121234


KEYWORD

nonn,more


AUTHOR

Dan Dima, Aug 21 2006


EXTENSIONS

Edited by R. J. Mathar, Oct 01 2008
a(7) from R. H. Hardin, Jun 19 2012. This makes it clear that the old A122527 was really a badlydescribed version of this sequence, and that a(7) was earlier found by Balakrishnan (bvarada2(AT)jhu.edu), Sep 17 2006.  N. J. A. Sloane, Jun 19 2012
Entry revised by N. J. A. Sloane, Jun 19 2012


STATUS

approved



