OFFSET
0,2
COMMENTS
Hankel transform of A002426(n+1). - Paul Barry, Mar 15 2008
Hankel transform of A007971(n+1). - Paul Barry, Sep 30 2009
Hankel transform of A103970 is a(n)/4^C(n+1,2). - Paul Barry, Nov 20 2009
The real part of Q^(n+1), where Q is the quaternion 1+i+j+k. - Stanislav Sykora, Jun 11 2012.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-4).
FORMULA
a(n) = A138340(n)/2^n. - Philippe Deléham, Nov 14 2008
a(n) = 2^(n+1)*cos(Pi*(n+1)/3). - Richard Choulet, Nov 19 2008
From Paul Barry, Oct 21 2009: (Start)
a(n) = Sum_{k=0..floor((n+1)/2)} C(n+1,2*k)*(-3)^k.
a(n) = ((1+i*sqrt(3))^(n+1) + (1-i*sqrt(3))^(n+1))/2, i=sqrt(-1). (End)
G.f.: G(0)/(2*x)-1/x, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 2^n*A057079(n+2). - R. J. Mathar, Mar 04 2018
Sum_{n>=0} 1/a(n) = 1/3. - Amiram Eldar, Feb 14 2023
MATHEMATICA
CoefficientList[Series[(1 - 4*x)/(1 - 2*x + 4*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, -4}, {1, -2}, 50] (* G. C. Greubel, Feb 28 2017 *)
PROG
(PARI) x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ G. C. Greubel, Feb 28 2017
CROSSREFS
KEYWORD
easy,sign,changed
AUTHOR
Paul Barry, Feb 11 2007
STATUS
approved