A008647 Expansion of g.f.: (1+x^9)/((1-x^4)*(1-x^6)).

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 3, 5, 4, 4, 4, 5, 4, 5, 4, 5, 5, 5, 4, 6, 5, 5, 5, 6, 5, 6, 5, 6, 6, 6, 5, 7, 6, 6, 6, 7, 6

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Molien series of binary octahedral group of order 48. Also Molien series for W_1 - W_3 of shadow of singly-even binary self-dual code.

References

• T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Eiichi Bannai, Etsuko Bannai, Michio Ozeki and Yasuo Teranishi, On the ring of simultaneous invariants for the Gleason-MacWilliams group, European J. Combin. 20 (1999), 619-627.
• J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1334.
• E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).
• Index entries for Molien series

Programs

GAP

a:=[1,0,0,0,1,0,1];; for n in [8..80] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 06 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^9)/((1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 06 2019

Maple

g:= proc(n) local m, r; m:= iquo(n, 12, 'r'); irem(r+1,2) *(m+1) -`if`(r=2, 1, 0) end: a:= n-> g(n) +`if`(n>8, g(n-9), 0); seq(a(n), n=0..100); # Alois P. Heinz, Oct 06 2008

Mathematica

CoefficientList[Series[(1+x^9)/((1-x^4)*(1-x^6)),{x,0,80}],x] (* or *) LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,0,1,0,1}, 80] (* Harvey P. Dale, Oct 10 2011 *)

PARI

a(n)=(9*(-1)^n+2*(n+7)+6*(-1)^(n\2))\24 \\ Charles R Greathouse IV, Feb 10 2017

Sage

def A008647_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^9)/((1-x^4)*(1-x^6))).list()
A008647_list(80) # G. C. Greubel, Sep 06 2019

Formula

G.f.: (1 - x^3 + x^6) / ( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^2 ).

G.f.: (1+x^6+x^9+x^15)/((1-x^4)*(1-x^12)).

It appears that the first differences have period 12. Hence in blocks of 12, the sequence is {1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0}+k for k=0,1,2,... - T. D. Noe, May 23 2008

a(n) = (6*A057077(n) +8*A057078(n) +1 +2*n +9*(-1)^n)/24. - R. J. Mathar, Jun 28 2009

a(n) = a(n-3) + a(n-4) - a(n-7), a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=0, a(6)=1. - Harvey P. Dale, Oct 10 2011

a(n) = floor((9*(-1)^n+2*(n+7)+6*(-1)^floor(n/2))/24). - Tani Akinari, Jun 17 2013

a(n) = floor(n/2) + floor(n/3) + floor(n/4) - n + 1. - Ridouane Oudra, Mar 21 2021