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A078029 Expansion of (1-x)/(1-2*x^3).

%I A078029 #26 Aug 03 2024 11:33:09
%S A078029 1,-1,0,2,-2,0,4,-4,0,8,-8,0,16,-16,0,32,-32,0,64,-64,0,128,-128,0,
%T A078029 256,-256,0,512,-512,0,1024,-1024,0,2048,-2048,0,4096,-4096,0,8192,
%U A078029 -8192,0,16384,-16384,0,32768,-32768,0,65536,-65536,0,131072,-131072,0,262144,-262144,0,524288,-524288,0
%N A078029 Expansion of (1-x)/(1-2*x^3).
%H A078029 G. C. Greubel, <a href="/A078029/b078029.txt">Table of n, a(n) for n = 0..1000</a>
%H A078029 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2).
%F A078029 G.f.: (1-x)/(1-2*x^3).
%F A078029 a(n) = (-1)^floor(4n/3)*(2-2*0^mod(n+1,3))^floor((n+1)/3). - _Wesley Ivan Hurt_, May 09 2015
%F A078029 a(n) = A077958(n) - A077958(n-1). - _R. J. Mathar_, Mar 04 2018
%F A078029 a(n) = (4^(n/6)/6)*(2 - 2^(2/3) + 2^(5/3)*sin(Pi*(2*n/3 + 5/6)) - 4*sin(Pi*(2*n/3 + 3/2))). - _Eric Simon Jacob_, Jul 14 2024
%p A078029 seq(op([2^n,-2^n,0]), n=0..60); # _Robert Israel_, May 11 2015
%t A078029 CoefficientList[Series[(1-x)/(1-2*x^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, May 10 2015 *)
%o A078029 (Magma) &cat[[2^n, -2^n, 0]: n in [0..60]]; // _Vincenzo Librandi_, May 10 2015
%o A078029 (PARI) my(x='x+O('x^60)); Vec((1-x)/(1-2*x^3)) \\ _G. C. Greubel_, Aug 05 2019
%o A078029 (Sage) ((1-x)/(1-2*x^3)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 05 2019
%o A078029 (GAP) a:=[1,-1,0];; for n in [4..60] do a[n]:=2*a[n-3]; od; a; # _G. C. Greubel_, Aug 05 2019
%K A078029 sign,easy
%O A078029 0,4
%A A078029 _N. J. A. Sloane_, Nov 17 2002