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%I A008647 #50 Jun 28 2023 20:58:49
%S A008647 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,
%T A008647 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,
%U A008647 6,6,6,5,7,6,6,6,7,6
%N A008647 Expansion of g.f.: (1+x^9)/((1-x^4)*(1-x^6)).
%C A008647 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.
%D A008647 T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.
%H A008647 T. D. Noe, <a href="/A008647/b008647.txt">Table of n, a(n) for n = 0..1000</a>
%H A008647 Eiichi Bannai, Etsuko Bannai, Michio Ozeki and Yasuo Teranishi, <a href="http://dx.doi.org/10.1006/eujc.1999.0310">On the ring of simultaneous invariants for the Gleason-MacWilliams group</a>, European J. Combin. 20 (1999), 619-627.
%H A008647 J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1109/18.59931">A new upper bound on the minimal distance of self-dual codes</a>, IEEE Trans. Inform. Theory, 36 (1990), 1319-1334.
%H A008647 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (<a href="http://neilsloane.com/doc/self.txt">Abstract</a>, <a href="http://neilsloane.com/doc/self.pdf">pdf</a>, <a href="http://neilsloane.com/doc/self.ps">ps</a>).
%H A008647 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,0,-1).
%H A008647 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F A008647 G.f.: ( 1 - x^3 + x^6) / ( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^2 ).
%F A008647 G.f.: (1+x^18)/((1-x^8)*(1-x^12) = (1+x^6+x^9+x^15)/((1-x^4)*(1-x^12)).
%F A008647 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
%F A008647 a(n) = (6*A057077(n) +8*A057078(n) +1 +2*n +9*(-1)^n)/24. - _R. J. Mathar_, Jun 28 2009
%F A008647 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
%F A008647 a(n) = floor((9*(-1)^n+2*(n+7)+6*(-1)^floor(n/2))/24). - _Tani Akinari_, Jun 17 2013
%F A008647 a(n) = floor(n/2) + floor(n/3) + floor(n/4) - n + 1. - _Ridouane Oudra_, Mar 21 2021
%p A008647 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
%t A008647 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 *)
%o A008647 (PARI) a(n)=(9*(-1)^n+2*(n+7)+6*(-1)^(n\2))\24 \\ _Charles R Greathouse IV_, Feb 10 2017
%o A008647 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1+x^9)/((1-x^4)*(1-x^6)) )); // _G. C. Greubel_, Sep 06 2019
%o A008647 (Sage)
%o A008647 def A008647_list(prec):
%o A008647     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008647     return P((1+x^9)/((1-x^4)*(1-x^6))).list()
%o A008647 A008647_list(80) # _G. C. Greubel_, Sep 06 2019
%o A008647 (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
%K A008647 nonn,easy,nice
%O A008647 0,13
%A A008647 _N. J. A. Sloane_

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