This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A077972 #23 Apr 22 2025 14:06:51
%S A077972 1,-1,2,-5,9,-18,37,-73,146,-293,585,-1170,2341,-4681,9362,-18725,
%T A077972 37449,-74898,149797,-299593,599186,-1198373,2396745,-4793490,9586981,
%U A077972 -19173961,38347922,-76695845,153391689,-306783378,613566757,-1227133513,2454267026,-4908534053,9817068105
%N A077972 Expansion of 1/(1+x-x^2+2*x^3).
%H A077972 G. C. Greubel, <a href="/A077972/b077972.txt">Table of n, a(n) for n = 0..1000</a>
%H A077972 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1,-2).
%F A077972 a(n) = (-1)^n * A077947(n).
%F A077972 G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+1 - x + 2*x^2 )/( x*(4*k+3 - x + 2*x^2 ) - 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 09 2013
%t A077972 LinearRecurrence[{-1,1,-2}, {1,-1,2}, 40] (* or *) CoefficientList[ Series[1/(1+x-x^2+2*x^3), {x,0,40}], x] (* _G. C. Greubel_, Jun 24 2019 *)
%o A077972 (PARI) Vec(1/(1+x-x^2+2*x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o A077972 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1+x-x^2+2*x^3) )); // _G. C. Greubel_, Jun 24 2019
%o A077972 (Sage) (1/(1+x-x^2+2*x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 24 2019
%o A077972 (GAP) a:=[1,-1,2];; for n in [4..40] do a[n]:=-a[n-1]+a[n-2]-2*a[n-3]; od; a; # _G. C. Greubel_, Jun 24 2019
%o A077972 (Python)
%o A077972 def A077972(n): return -(((4<<n)|3)//7) if n&1 else ((4<<n)|3)//7 # _Chai Wah Wu_, Apr 22 2025
%Y A077972 Cf. A077947.
%K A077972 sign,easy
%O A077972 0,3
%A A077972 _N. J. A. Sloane_, Nov 17 2002