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

0,2

Comments

Comment from N. J. A. Sloane, May 28 2026 (Start):

Consider the square spiral with cells numbered starting at 0, as in A392177 and A392178. The sequence lists the cells that can be reached in a single move by a queen placed at cell 0. A396179 lists the cells that the queen cannot reach in a single move, and begins with the eight cells that could be reached by a knight placed at cell 0.

If an initial 0 is prepended, the present sequence is the union of A001107, A002939, A002943, A007742, A016742, A033954, A033991, and A033996 (compare A118729). (End)

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).

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

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 *)
(* Alternative: *)
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 *)

Prog

(PARI) my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^8))) \\ G. C. Greubel, Sep 09 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^8)) )); // G. C. Greubel, Sep 09 2019
(SageMath)
def A008726_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^8))).list()
A008726_list(80) # G. C. Greubel, Sep 09 2019
(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
(Python)
def A008726(n): return ((n>>3)+1)*(n+1-(n>>1&-4)) # Chai Wah Wu, May 28 2026

Crossrefs

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

Cf. also A001107, A002939, A002943, A007742, A016742, A033954, A033991, and A033996.

See also A118729, A392177, A392178, A396179.

Sequence in context: A122936 A330729 A118729 * A302834 A022788 A141340

Adjacent sequences: A008723 A008724 A008725 * A008727 A008728 A008729

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Minor edits by Jon E. Schoenfield, Mar 28 2014

Status

approved