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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Nov 03 2012
STATUS
approved