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.

1, 2, 11, 172, 6327, 474286, 67147431, 17080038508

Offset: 0

Author

Dan Dima, Aug 21 2006

Keywords

• nonn
• more

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 n-th 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

• 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.

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 badly-described 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