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A106731 Expansion of -2*x/(1 - 4*x + 2*x^2). 2

%I

%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 Previous name was: First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-2],[1,4]] and v is the column vector [0,1].

%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>

%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 a(n) = -(1/2)*sqrt(2) * ((2+sqrt(2))^n - (2-sqrt(2))^n). - _Paolo P. Lava_, Oct 07 2008

%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..27);

%t M = {{0, -2}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]

%t CoefficientList[Series[-2 x/(1 - 4 x + 2 x^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 04 2013 *)

%Y Cf. A060995.

%Y Equals -2*A007070(n-1), n>=1.

%K sign

%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

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Last modified March 24 06:56 EDT 2019. Contains 321444 sequences. (Running on oeis4.)