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

0,2

Comments

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].

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).

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.

a(n) = -(1/2)*sqrt(2) * ((2+sqrt(2))^n - (2-sqrt(2))^n). - Paolo P. Lava, Oct 07 2008

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A000129, A002203, A007070, A060995.

Sequence in context: A279193 A280279 A060995 * A318010 A291383 A277653

Adjacent sequences: A106728 A106729 A106730 * A106732 A106733 A106734

Keyword

sign,easy,less

Author

Roger L. Bagula, May 30 2005

Extensions

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

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

Status

approved