[seqfan] Re: [math-fun] Re: EXACT matrix factorizations

[Victor Miller]

I developed a method of pushing this sequence much further.  I can now
calculate through a(25).  The idea is the following:

whether or not B is the square of a matrix is solely a function of its
conjugacy class.  So I look at all possible Jordan canonical forms for an n
by n matrix, and determine which ones of these can occur in the square of a
matrix.  After enumerating these, I calculate the cardinality of the
centralizer of each, which, divided into the cardinality of the general
linear group of invertible matrices, gives the number of such.  In order to
avoid combinatorial explosion, I actually look at multisets of such
matrices, and then multiply them by the appropriate multinomial coefficient
corresponding to choices of distinct (actually non-conjugate) eigenvalues
of a given degree above GF(2).  I also have a very hand-wavy proof that

lim log_2 (a(n))/n^2 = 1.  That is the the set of matrices which aren't
squares is a negligible proportion.  In the process of doing this I also
found two sequences which aren't in OEIS (even with superseeker).

Victor


On Wed, May 8, 2013 at 5:48 AM, Giovanni Resta <g.resta at iit.cnr.it> wrote:

>
>  a(n) = number of squares in M(n,2) =
>>> ring of nxn matrices over GF(2),
>>> beginning with n = 1:
>>>                      2,10,260,31096
>>> which is not in the OEIS.  Perhaps some interested soul can extend this.
>>>
>>
> I've added a(5) = 13711952 and in about 3.5 hours I can add a(6).
>
> Giovanni
>
>
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