Search: id:A121231
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%I A121231
%S A121231 2,11,172,6327,474286,67147431
%N A121231 Number of n X n binary matrices M such that M^2 is also a binary matrix.
%C A121231 A binary matrix is a real matrix with entries 0 and 1.
%C A121231 Comments from Brendan McKay (bdm(AT)cs.anu.edu.au), 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.
%C A121231 Comment Rob Pratt (Rob.Pratt(AT)sas.com), 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:
%C A121231 0 0 1 0 0
%C A121231 0 0 0 0 1
%C A121231 1 1 0 0 1
%C A121231 1 1 0 1 0
%C A121231 1 1 0 1 0.
%C A121231 The optimal values seem to match A070214, verified for n<=7.
%C A121231 Term (5,1) of n-th power of the 5x5 matrix shown = A001045(n), the Jacobsthal 
               sequence. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 03 2008]
%H A121231 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GraphPower.html">Background information about adjacency matrices</
               a>
%H A121231 Wikipedia, <a href="http://en.wikipedia.org/wiki/Adjacency_matrix">Background 
               information about adjacency matrices</a>
%Y A121231 Cf. A121294.
%Y A121231 A001045 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 03 2008]
%Y A121231 Sequence in context: A051255 A120445 A003088 this_sequence A122527 A039747 
               A049531
%Y A121231 Adjacent sequences: A121228 A121229 A121230 this_sequence A121232 A121233 
               A121234
%K A121231 nonn
%O A121231 1,1
%A A121231 Dan Dima (dimad72(AT)gmail.com), Aug 21 2006
%E A121231 Edited by R. J. Mathar, Oct 01 2008

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