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A274313 The number of conjugacy classes of n X n matrices over GF(2) which are squares of other such matrices.

%I A274313 #45 Nov 19 2024 00:41:55
%S A274313 1,2,4,10,22,46,96,198,406,826,1668,3362,6770,13590,27248,54614,
%T A274313 109378,218946,438180,876738,1753998,3508726,7018368,14038006,
%U A274313 28077846,56157954,112318900,224642090,449289666,898586438,1797182704,3594378014,7188772666,14377567834,28755164100,57510365698,115020782350,230041628622,460083340304,920166792942
%N A274313 The number of conjugacy classes of n X n matrices over GF(2) which are squares of other such matrices.
%H A274313 Andrew Howroyd, <a href="/A274313/b274313.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..60 from N. J. A. Sloane)
%H A274313 Victor S. Miller, <a href="http://arxiv.org/abs/1606.09299">Counting Matrices that are Squares</a>, arXiv:1606.09299 [math.GR], 2016.
%H A274313 <a href="/index/Mat#binmat">Index entries for matrices, binary, which are squares</a>
%F A274313 G.f.: Product_{n>=1} (1-2*z^(2*n))/((1-2*z^n)*(1-2*z^(4*n))). - _Jean-François Alcover_, Dec 12 2018, after Victor S. Miller.
%p A274313 seq(coeff(series(mul((1-2*x^(2*k))/((1-2*x^k)*(1-2*x^(4*k))), k=1..n),x,n+1), x, n), n = 0 .. 40); # _Muniru A Asiru_, Dec 13 2018
%t A274313 terms = 40;
%t A274313 Product[(1-2z^(2n))/(1-2z^n)/(1-2z^(4n)), {n, 1, terms}] + O[z]^terms // CoefficientList[#, z]& (* _Jean-François Alcover_, Dec 12 2018 *)
%o A274313 (PARI) seq(n)=Vec(prod(i=1, n, (1-2*x^(2*i))/((1-2*x^i)*(1-2*x^(4*i)) + O(x*x^n)))) \\ _Andrew Howroyd_, Dec 12 2018
%o A274313 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-2*x^(2*k))/((1-2*x^k)*(1-2*x^(4*k))): k in [1..m/2]]))); // _G. C. Greubel_, Dec 16 2018
%o A274313 (Sage) m=40; s=(prod((1-2*x^(2*k))/((1-2*x^k)*(1-2*x^(4*k))) for k in (1..m/2))).series(x, m); s.coefficients(x, sparse=False) # _G. C. Greubel_, Dec 16 2018
%Y A274313 Cf. A266462.
%K A274313 nonn
%O A274313 0,2
%A A274313 _N. J. A. Sloane_, Jun 25 2016