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A121231
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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.
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6
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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E. W. Weisstein, (0,1)-Matrix, MathWorld. [P. Petsie, Aug 03 2008]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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