A006952 Number of conjugacy classes in GL(n,3). (Formerly M1842)

1, 2, 8, 24, 78, 232, 720, 2152, 6528, 19578, 58944, 176808, 531128, 1593288, 4781952, 14345792, 43043622, 129130584, 387411144, 1162232520, 3486755688, 10460266224, 31380972784, 94142915640, 282429275616, 847287817866, 2541865038832, 7625595108432

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. D. Smith, personal communication.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700

W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 162

I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).

Formula

G.f.: Product_{n>=1} (1-x^n)/(1-3*x^n). - Joerg Arndt, Jan 02 2013

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

a(n) ~ 3^n - (1+sqrt(3) + (-1)^n*(1-sqrt(3))) * 3^(n/2) / 4. - Vaclav Kotesovec, May 06 2018

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(3^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

Maple

with(numtheory):
b:= n-> add(phi(d)*3^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

Mathematica

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 *)

Prog

(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
(PARI)
N=66; x='x+O('x^N);
gf=prod(n=1, N, (1-x^n)/(1-3*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */

Crossrefs

Cf. A006951, A049314, A049315, A049316, A304082.

Sequence in context: A026070 A093833 A228404 * A327550 A034741 A063727

Adjacent sequences: A006949 A006950 A006951 * A006953 A006954 A006955

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Alois P. Heinz, Nov 03 2012

Status

approved