

A106731


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


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
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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



FORMULA

G.f.: 2*x/(14*x+2*x^2).
a(n) = 4*a(n1)  2*a(n2); a(0)=0, a(1)=2.
a(n) = (1/2)*sqrt(2) * ((2+sqrt(2))^n  (2sqrt(2))^n).  Paolo P. Lava, Oct 07 2008
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[n1]2*a[n2] od: seq(a[n], n=0..30);


MATHEMATICA

M= {{0, 2}, {1, 4}}; v[1]= {0, 1}; v[n_]:= v[n]= M.v[n1]; 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(n1)  2*Self(n2): 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^((n1)/2)*lucas_number2(n, 2, 1)


CROSSREFS



KEYWORD

sign,easy,less


AUTHOR



EXTENSIONS

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


STATUS

approved



