OFFSET
0,2
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).
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
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
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