A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2).

1, 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

0,3

Comments

In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Links

Michael De Vlieger, Table of n, a(n) for n = 0..3322

Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.

Index entries for linear recurrences with constant coefficients, signature (2,-4).

Formula

From Philippe Deléham, Nov 14 2008: (Start)

a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)

a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008

a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008

a(n) = -8*a(n-3). - Paul Curtz, Apr 22 2011

From Sergei N. Gladkovskii, Jul 27 2012: (Start)

G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).

E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

a(n) = A088138(n+1) - A088138(n). - R. J. Mathar, Mar 04 2018

a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019

a(n) = 2^(n-1)*A087204(n). - G. C. Greubel, Feb 11 2023

Sum_{n>=0} 1/a(n) = 4/3. - Amiram Eldar, Feb 14 2023

Mathematica

CoefficientList[Series[(1-x)/(1-2x+4x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, -4}, {1, 1}, 30] (* Harvey P. Dale, Nov 11 2014 *)

Prog

(Magma) [2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023
(SageMath) [2^n*chebyshev_T(n, 1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023

Crossrefs

Cf. A087204, A088138, A098158, A104537, A124182, A128018.

Sequence in context: A290378 A104537 A128018 * A019240 A269510 A093907

Adjacent sequences: A138227 A138228 A138229 * A138231 A138232 A138233

Keyword

easy,sign

Author

Paul Barry, Mar 06 2008

Status

approved