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Expansion of (1-4*x)/(1-2*x+4*x^2).
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%I #33 Sep 20 2024 03:20:07

%S 1,-2,-8,-8,16,64,64,-128,-512,-512,1024,4096,4096,-8192,-32768,

%T -32768,65536,262144,262144,-524288,-2097152,-2097152,4194304,

%U 16777216,16777216,-33554432,-134217728,-134217728,268435456,1073741824,1073741824,-2147483648,-8589934592

%N Expansion of (1-4*x)/(1-2*x+4*x^2).

%C Hankel transform of A128014(n+1). Binomial transform of A128019.

%C Hankel transform of A002426(n+1). - _Paul Barry_, Mar 15 2008

%C Hankel transform of A007971(n+1). - _Paul Barry_, Sep 30 2009

%C Hankel transform of A103970 is a(n)/4^C(n+1,2). - _Paul Barry_, Nov 20 2009

%C The real part of Q^(n+1), where Q is the quaternion 1+i+j+k. - Stanislav Sykora, Jun 11 2012.

%H G. C. Greubel, <a href="/A128018/b128018.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-4).

%F a(n) = A138340(n)/2^n. - _Philippe Deléham_, Nov 14 2008

%F a(n) = 2^(n+1)*cos(Pi*(n+1)/3). - _Richard Choulet_, Nov 19 2008

%F From _Paul Barry_, Oct 21 2009: (Start)

%F a(n) = Sum_{k=0..floor((n+1)/2)} C(n+1,2*k)*(-3)^k.

%F a(n) = ((1+i*sqrt(3))^(n+1) + (1-i*sqrt(3))^(n+1))/2, i=sqrt(-1). (End)

%F 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 a(n) = 2^n*A057079(n+2). - _R. J. Mathar_, Mar 04 2018

%F Sum_{n>=0} 1/a(n) = 1/3. - _Amiram Eldar_, Feb 14 2023

%t 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 (PARI) x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ _G. C. Greubel_, Feb 28 2017

%Y Cf. A002426, A007971, A128014, A103970, A128019, A138340.

%K easy,sign

%O 0,2

%A _Paul Barry_, Feb 11 2007