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

%I A106731 #41 May 21 2024 02:29:27
%S A106731 0,-2,-8,-28,-96,-328,-1120,-3824,-13056,-44576,-152192,-519616,
%T A106731 -1774080,-6057088,-20680192,-70606592,-241065984,-823050752,
%U A106731 -2810071040,-9594182656,-32756588544,-111837988864,-381838778368,-1303679135744,-4451038986240,-15196797673472
%N A106731 Expansion of -2*x/(1 - 4*x + 2*x^2).
%C A106731 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 A106731 Vincenzo Librandi, <a href="/A106731/b106731.txt">Table of n, a(n) for n = 0..300</a>
%H A106731 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F A106731 G.f.: -2*x/(1-4*x+2*x^2).
%F A106731 a(n) = -2*A007070(n-1) for n>=1.
%F A106731 a(n) = 4*a(n-1) - 2*a(n-2); a(0)=0, a(1)=-2.
%F A106731 From _G. C. Greubel_, Sep 10 2021: (Start)
%F A106731 a(2*n) = -2^(n+1)*Pell(2*n) = -2^(n+1)*A000129(2*n).
%F A106731 a(2*n+1) = -2^n*Q(2n+1) = -2^n*A002203(2*n+1). (End)
%F A106731 E.g.f.: -sqrt(2)*exp(2*x)*sinh(sqrt(2)*x). - _Stefano Spezia_, May 20 2024
%p A106731 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 A106731 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 A106731 CoefficientList[Series[-2x/(1 -4x +2x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 04 2013 *)
%o A106731 (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 A106731 (Sage)
%o A106731 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 A106731 [a(n) for n in (0..30)] # _G. C. Greubel_, Sep 10 2021
%Y A106731 Cf. A000129, A002203, A007070, A060995.
%K A106731 sign,easy,less
%O A106731 0,2
%A A106731 _Roger L. Bagula_, May 30 2005
%E A106731 Edited by _N. J. A. Sloane_, Apr 30 2006
%E A106731 Further editing and simpler name, _Joerg Arndt_, Oct 02 2013