A128018 Expansion of (1-4*x)/(1-2*x+4*x^2).

1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset

0,2

Comments

Hankel transform of A128014(n+1). Binomial transform of A128019.

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) = 2^n*(cos(pi*n/3) - sqrt(3)*sin(pi*n)/3).

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+1,2k)*(-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

Cf. A002426, A007971, A128014, A103970, A128019, A138340.

Sequence in context: A281914 A290378 A104537 * A138230 A019240 A269510

Adjacent sequences: A128015 A128016 A128017 * A128019 A128020 A128021

Keyword

easy,sign

Author

Paul Barry, Feb 11 2007

Status

approved