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%I M1842 #50 Sep 08 2022 08:44:35
%S 1,2,8,24,78,232,720,2152,6528,19578,58944,176808,531128,1593288,
%T 4781952,14345792,43043622,129130584,387411144,1162232520,3486755688,
%U 10460266224,31380972784,94142915640,282429275616,847287817866,2541865038832,7625595108432
%N Number of conjugacy classes in GL(n,3).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D W. D. Smith, personal communication.
%H Alois P. Heinz, <a href="/A006952/b006952.txt">Table of n, a(n) for n = 0..700</a>
%H W. Feit and N. J. Fine, <a href="https://projecteuclid.org/euclid.dmj/1077468920">Pairs of commuting matrices over a finite field</a>, Duke Math. Journal, 27 (1960) 91-94.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=162">Encyclopedia of Combinatorial Structures 162</a>
%H I. G. Macdonald, <a href="https://doi.org/10.1017/S0004972700006882">Numbers of conjugacy classes in some finite classical groups</a>, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).
%F G.f.: Product_{n>=1} (1-x^n)/(1-3*x^n). - _Joerg Arndt_, Jan 02 2013
%F The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in Product_{k>=1} (1-t^k)/(1-q*t^k). - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
%F a(n) ~ 3^n - (1+sqrt(3) + (-1)^n*(1-sqrt(3))) * 3^(n/2) / 4. - _Vaclav Kotesovec_, May 06 2018
%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(3^(k/d) - 1) ) * x^k/k). - _Ilya Gutkovskiy_, Sep 27 2018
%p with(numtheory):
%p b:= n-> add(phi(d)*3^(n/d), d=divisors(n))/n-1:
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 03 2012
%t b[n_] := Sum[EulerPhi[d]*3^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 17 2014, after _Alois P. Heinz_ *)
%o (Magma) /* The program does not work for n>12: */ [1] cat [NumberOfClasses(GL(n, 3)) : n in [1..12]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by _Vincenzo Librandi_, Jan 23 2013
%o (PARI)
%o N=66; x='x+O('x^N);
%o gf=prod(n=1, N, (1-x^n)/(1-3*x^n) );
%o v=Vec(gf)
%o /* _Joerg Arndt_, Jan 02 2013 */
%Y Cf. A006951, A049314, A049315, A049316, A304082.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Alois P. Heinz_, Nov 03 2012
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