A008726 Molien series 1/((1-x)^2*(1-x^8)) for 3-dimensional group [2,n] = *22n.

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168, 175, 182, 189, 196, 203, 210, 217, 224, 232, 240, 248, 256, 264, 272, 280

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 191
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A001840, A001972, A008724, A008725, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Programs

GAP

a:=[1,2,3,4,5,6,7,8,10,12];; for n in [11..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^8)) )); // G. C. Greubel, Sep 09 2019

Maple

seq(coeff(series(1/(1-x)^2/(1-x^8), x, n+1), x, n), n=0..80);

Mathematica

CoefficientList[Series[1/((1-x)^2*(1-x^8)), {x,0,80}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,10,12}, 80] (* Harvey P. Dale, Jan 07 2015 *)

PARI

my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^8))) \\ G. C. Greubel, Sep 09 2019

Sage

def A008726_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^8))).list()
A008726_list(80) # G. C. Greubel, Sep 09 2019

Formula

G.f.: 1/((1-x)^2*(1-x^8)).

From Mitch Harris, Sep 08 2008: (Start)

a(n) = Sum_{j=0..n+8} floor(j/8).

a(n-8) = (1/2)*floor(n/8)*(2*n-6-8*floor(n/8)). (End)

a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - R. J. Mathar, Apr 20 2010

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Minor edits by Jon E. Schoenfield, Mar 28 2014