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%I #34 Sep 08 2022 08:45:13
%S 1,-2,8,-16,64,-128,512,-1024,4096,-8192,32768,-65536,262144,-524288,
%T 2097152,-4194304,16777216,-33554432,134217728,-268435456,1073741824,
%U -2147483648,8589934592,-17179869184,68719476736,-137438953472
%N Expansion of (1-2*x)/(1-8*x^2).
%C Second inverse binomial transform of A094013. Third inverse binomial transform of A000129(2n-1).
%C The unsigned sequence has g.f. (1+2*x)/(1-8*x^2) and abs(a(n)) = 2^(3*n/2)*(1/2 + sqrt(2)/4 + (1/2 - sqrt(2)/4)*(-1)^n).
%H G. C. Greubel, <a href="/A094014/b094014.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 (0,8).
%F a(n) = (2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (-2*sqrt(2))^n*(1/2 + sqrt(2)/4).
%F a(n) = (-2)^n * A016116(n). - _R. J. Mathar_, Apr 28 2008
%F Abs(a(n)) = A113836(n+1) - A113836(n) for n > 0. - _Reinhard Zumkeller_, Feb 22 2010
%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011
%F a(n) = Sum_{k=0..n} A158020(n,k)*3^k. - _Philippe Deléham_, Dec 01 2011
%F E.g.f.: cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x). - _G. C. Greubel_, Dec 04 2021
%t LinearRecurrence[{0,8}, {1,-2}, 40] (* _G. C. Greubel_, Dec 04 2021 *)
%o (Magma) [n le 2 select (-2)^(n-1) else 8*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Dec 04 2021
%o (Sage) [(-2)^n*2^(n//2) for n in (0..40)] # _G. C. Greubel_, Dec 04 2021
%Y Cf. A000129, A016116, A094013, A113836, A158020.
%K easy,sign
%O 0,2
%A _Paul Barry_, Apr 21 2004