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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,-4).
Programs
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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 *) -
PARI
x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ G. C. Greubel, Feb 28 2017
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
Comments