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A006952
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Number of conjugacy classes in GL(n,3).
(Formerly M1842)
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25
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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
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. D. Smith, personal communication.
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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