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

0, -2, -8, -28, -96, -328, -1120, -3824, -13056, -44576, -152192, -519616, -1774080, -6057088, -20680192, -70606592, -241065984, -823050752, -2810071040, -9594182656, -32756588544, -111837988864, -381838778368, -1303679135744, -4451038986240, -15196797673472

Offset: 0

Author

Roger L. Bagula, May 30 2005

Keywords

• sign
• easy
• less

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A000129, A002203, A007070, A060995.

Programs

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

Maple

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);

Mathematica

M= {{0,-2}, {1,4}}; v[1]= {0,1}; v[n_]:= v[n]= M.v[n-1]; Table[Abs[v[n][[1]]], {n, 30}]
CoefficientList[Series[-2x/(1 -4x +2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2013 *)

Sage

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)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 10 2021

Formula

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

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

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

From G. C. Greubel, Sep 10 2021: (Start)

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

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

E.g.f.: -sqrt(2)*exp(2*x)*sinh(sqrt(2)*x). - Stefano Spezia, May 20 2024

Extensions

Edited by N. J. A. Sloane, Apr 30 2006

Further editing and simpler name, Joerg Arndt, Oct 02 2013