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

%I #41 May 21 2024 02:29:27

%S 0,-2,-8,-28,-96,-328,-1120,-3824,-13056,-44576,-152192,-519616,

%T -1774080,-6057088,-20680192,-70606592,-241065984,-823050752,

%U -2810071040,-9594182656,-32756588544,-111837988864,-381838778368,-1303679135744,-4451038986240,-15196797673472

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

%C See a Oct 01 2013 comment on A007070 where it is pointed out that this sequence, interspersed with zeros, appears, together with A007070, also interspersed with zeros, in the representation of nonnegative powers of the algebraic number rho(8) = 2*cos(Pi/8) in the power basis of the number field Q(rho(8)) of degree 4, known from the octagon. - _Wolfdieter Lang_, Oct 02 2013

%H Vincenzo Librandi, <a href="/A106731/b106731.txt">Table of n, a(n) for n = 0..300</a>

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

%F G.f.: -2*x/(1-4*x+2*x^2).

%F a(n) = -2*A007070(n-1) for n>=1.

%F a(n) = 4*a(n-1) - 2*a(n-2); a(0)=0, a(1)=-2.

%F From _G. C. Greubel_, Sep 10 2021: (Start)

%F a(2*n) = -2^(n+1)*Pell(2*n) = -2^(n+1)*A000129(2*n).

%F a(2*n+1) = -2^n*Q(2n+1) = -2^n*A002203(2*n+1). (End)

%F E.g.f.: -sqrt(2)*exp(2*x)*sinh(sqrt(2)*x). - _Stefano Spezia_, May 20 2024

%p a[0]:=0: a[1]:=-2: for n from 2 to 27 do a[n]:=4*a[n-1]-2*a[n-2] od: seq(a[n], n=0..30);

%t M= {{0,-2}, {1,4}}; v[1]= {0,1}; v[n_]:= v[n]= M.v[n-1]; Table[Abs[v[n][[1]]], {n, 30}]

%t CoefficientList[Series[-2x/(1 -4x +2x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 04 2013 *)

%o (Magma) [n le 2 select -(1+(-1)^n) else 4*Self(n-1) - 2*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Sep 10 2021

%o (Sage)

%o def a(n): return -2^((n+2)/2)*lucas_number1(n,2,-1) if (n%2==0) else -2^((n-1)/2)*lucas_number2(n,2,-1)

%o [a(n) for n in (0..30)] # _G. C. Greubel_, Sep 10 2021

%Y Cf. A000129, A002203, A007070, A060995.

%K sign,easy,less

%O 0,2

%A _Roger L. Bagula_, May 30 2005

%E Edited by _N. J. A. Sloane_, Apr 30 2006

%E Further editing and simpler name, _Joerg Arndt_, Oct 02 2013