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%I #50 Jan 01 2024 11:44:29
%S 1,1,-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-x)/(1 - 2*x + 4*x^2).
%C In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.
%C Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - _Philippe Deléham_, Dec 02 2008
%H Michael De Vlieger, <a href="/A138230/b138230.txt">Table of n, a(n) for n = 0..3322</a>
%H Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, <a href="https://doi.org/10.1007/s00006-019-0969-9">On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis</a>, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-4).
%F From _Philippe Deléham_, Nov 14 2008: (Start)
%F a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.
%F a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)
%F a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - _Philippe Deléham_, Nov 15 2008
%F a(n) = 2^n*cos(Pi*n/3). - _Richard Choulet_, Nov 19 2008
%F a(n) = -8*a(n-3). - _Paul Curtz_, Apr 22 2011
%F From _Sergei N. Gladkovskii_, Jul 27 2012: (Start)
%F G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).
%F E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F a(n) = A088138(n+1) - A088138(n). - _R. J. Mathar_, Mar 04 2018
%F a(n) = (-1)^n*A104537(n). - _R. J. Mathar_, May 21 2019
%F a(n) = 2^(n-1)*A087204(n). - _G. C. Greubel_, Feb 11 2023
%F Sum_{n>=0} 1/a(n) = 4/3. - _Amiram Eldar_, Feb 14 2023
%t CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* _Harvey P. Dale_, Nov 11 2014 *)
%o (Magma) [2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // _G. C. Greubel_, Feb 11 2023
%o (SageMath) [2^n*chebyshev_T(n,1/2) for n in range(31)] # _G. C. Greubel_, Feb 11 2023
%Y Cf. A087204, A088138, A098158, A104537, A124182, A128018.
%K easy,sign
%O 0,3
%A _Paul Barry_, Mar 06 2008
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