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The number of conjugacy classes of invertible n X n matrices over GF(2) which are squares of other such matrices.
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%I #33 Jan 28 2021 20:55:26

%S 1,1,2,5,10,20,41,82,166,334,667,1336,2682,5360,10724,21467,42936,

%T 85876,171786,343574,687184,1374427,2748852,5497766,10995706,21991402,

%U 43982908,87966150,175932383,351864964,703730584,1407461288,2814923196,5629847656,11259695532

%N The number of conjugacy classes of invertible n X n matrices over GF(2) which are squares of other such matrices.

%C It follows from the form of the generating function that a(n) is asymptotic to alpha*2^n where alpha = Product_{n>=1} (1-(1/16)^n)(1-2(1/2)^n)/ ((1-2(1/16)^n)(1-(1/4)^n))).

%H N. J. A. Sloane, <a href="/A266462/b266462.txt">Table of n, a(n) for n = 0..60</a> [Based on the table in Miller (2016)]

%H Victor S. Miller, <a href="http://arxiv.org/abs/1606.09299">Counting Matrices that are Squares</a>, arXiv:1606.09299 [math.GR], 2016.

%H <a href="/index/Mat#binmat">Index entries for matrices, binary, which are squares</a>

%F G.f.: Product_{n>=1} (1-2*x^(2*n))*(1-x^(2*n))/((1-2*x^n)*(1-2*x^(4*n))(1+x^(2*n-1))).

%t terms = 35; CoefficientList[Product[(1-2x^(2n))(1-x^(2n))/((1-2x^n) (1-2x^(4n))(1+x^(2n-1))), {n, 1, terms}] + O[x]^terms, x] (* _Jean-François Alcover_, Aug 06 2018 *)

%Y Cf. A006950.

%K nonn

%O 0,3

%A _Victor S. Miller_, Dec 29 2015

%E More terms from _Alois P. Heinz_, Dec 29 2015