A242587 The number of conjugacy classes of n X n matrices over F_3.

1, 3, 12, 39, 129, 399, 1245, 3783, 11514, 34734, 104754, 314922, 946623, 2842077, 8532147, 25603788, 76830033, 230513439, 691598901, 2074870002, 6224790639, 18674600664, 56024355396, 168073769199, 504222998115, 1512671142432, 4538018555652, 13614062210490

Offset: 0

Author

R. J. Mathar, May 18 2014

Keywords

• nonn

Comments

Apparently the Euler transform of A001867.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• K. E. Morrison, Integer sequences and matrices over finite fields, J. Int. Seq. 9 (2006) # 06.2.1, Section 1.16.

Crossrefs

Cf. A070933 (over F_2).

Column k=3 of A246935.

Programs

Maple

A242587 := proc(n)
    local r,x ;
    if n  = 0 then
        1;
    else
        1/mul(1-3*x^r,r=1..n) ;
        convert(%,parfrac,x) ;
        coeftayl(%,x=0,n) ;
    end if;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 3*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, 3*b[n-i, i]]]] ; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 03 2015, after Alois P. Heinz *)
(O[x]^20 - 2/QPochhammer[3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Maxima

S(n, m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

Formula

G.f.: 1/Product_{r>=1} (1-3*x^r).

a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, 3*S(n-k,k))+3, S(n,n)=3, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014

a(n) ~ c * 3^n, where c = Product_{k>=1} 1/(1-1/3^k) = 1.7853123419985341903674... . - Vaclav Kotesovec, Mar 19 2015

G.f.: Sum_{i>=0} 3^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018