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 A128018 #33 Sep 20 2024 03:20:07
%S A128018 1,-2,-8,-8,16,64,64,-128,-512,-512,1024,4096,4096,-8192,-32768,
%T A128018 -32768,65536,262144,262144,-524288,-2097152,-2097152,4194304,
%U A128018 16777216,16777216,-33554432,-134217728,-134217728,268435456,1073741824,1073741824,-2147483648,-8589934592
%N A128018 Expansion of (1-4*x)/(1-2*x+4*x^2).
%C A128018 Hankel transform of A128014(n+1). Binomial transform of A128019.
%C A128018 Hankel transform of A002426(n+1). - _Paul Barry_, Mar 15 2008
%C A128018 Hankel transform of A007971(n+1). - _Paul Barry_, Sep 30 2009
%C A128018 Hankel transform of A103970 is a(n)/4^C(n+1,2). - _Paul Barry_, Nov 20 2009
%C A128018 The real part of Q^(n+1), where Q is the quaternion 1+i+j+k. - Stanislav Sykora, Jun 11 2012.
%H A128018 G. C. Greubel, <a href="/A128018/b128018.txt">Table of n, a(n) for n = 0..1000</a>
%H A128018 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-4).
%F A128018 a(n) = A138340(n)/2^n. - _Philippe Deléham_, Nov 14 2008
%F A128018 a(n) = 2^(n+1)*cos(Pi*(n+1)/3). - _Richard Choulet_, Nov 19 2008
%F A128018 From _Paul Barry_, Oct 21 2009: (Start)
%F A128018 a(n) = Sum_{k=0..floor((n+1)/2)} C(n+1,2*k)*(-3)^k.
%F A128018 a(n) = ((1+i*sqrt(3))^(n+1) + (1-i*sqrt(3))^(n+1))/2, i=sqrt(-1). (End)
%F A128018 G.f.: G(0)/(2*x)-1/x, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013
%F A128018 a(n) = 2^n*A057079(n+2). - _R. J. Mathar_, Mar 04 2018
%F A128018 Sum_{n>=0} 1/a(n) = 1/3. - _Amiram Eldar_, Feb 14 2023
%t A128018 CoefficientList[Series[(1 - 4*x)/(1 - 2*x + 4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,-4},{1,-2},50] (* _G. C. Greubel_, Feb 28 2017 *)
%o A128018 (PARI) x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ _G. C. Greubel_, Feb 28 2017
%Y A128018 Cf. A002426, A007971, A128014, A103970, A128019, A138340.
%K A128018 easy,sign
%O A128018 0,2
%A A128018 _Paul Barry_, Feb 11 2007