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 A008647 Expansion of g.f.: (1+x^9)/((1-x^4)*(1-x^6)). 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 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 FORMULA G.f.: ( 1 - x^3 + x^6) / ( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^2 ). G.f.: (1+x^18)/((1-x^8)*(1-x^12) = (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 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 *) PROG (PARI) a(n)=(9*(-1)^n+2*(n+7)+6*(-1)^(n\2))\24 \\ Charles R Greathouse IV, Feb 10 2017 (Magma) R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^9)/((1-x^4)*(1-x^6)) )); // G. C. Greubel, Sep 06 2019 (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 (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 CROSSREFS Sequence in context: A335449 A318464 A051265 * A036475 A330746 A316555 Adjacent sequences: A008644 A008645 A008646 * A008648 A008649 A008650 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane STATUS approved

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Last modified December 3 07:48 EST 2023. Contains 367531 sequences. (Running on oeis4.)