%I #24 Sep 22 2025 16:00:41

%S 1,-1,1,-3,3,-5,9,-11,19,-29,41,-67,99,-149,233,-347,531,-813,1225,

%T -1875,2851,-4325,6601,-10027,15251,-23229,35305,-53731,81763,-124341,

%U 189225,-287867,437907,-666317,1013641,-1542131,2346275,-3569413,5430537,-8261963,12569363,-19123037,29093289

%N Expansion of (1-x)/(1-x^2+2*x^3).

%H G. C. Greubel, <a href="/A078028/b078028.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,-2).

%p seq(coeff(series((1-x)/(1-x^2+2*x^3), x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Aug 04 2019

%t CoefficientList[Series[(1-x)/(1-x^2+2*x^3), {x,0,60}], x] (* _G. C. Greubel_, Aug 04 2019 *)

%o (PARI) Vec((1-x)/(1-x^2+2*x^3)+O(x^60)) \\ _Charles R Greathouse IV_, Sep 25 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1-x^2+2*x^3) )); // _G. C. Greubel_, Aug 04 2019

%o (SageMath) ((1-x)/(1-x^2+2*x^3)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 04 2019

%o (GAP) a:=[1,-1,1];; for n in [4..60] do a[n]:=a[n-2]-2*a[n-3]; od; a; # _G. C. Greubel_, Aug 04 2019

%K sign,easy

%O 0,4

%A _N. J. A. Sloane_, Nov 17 2002