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A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2). 14
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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
Sequence in context: A290378 A104537 A128018 * A019240 A269510 A093907
KEYWORD
easy,sign
AUTHOR
Paul Barry, Mar 06 2008
STATUS
approved

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Last modified April 25 09:49 EDT 2024. Contains 371967 sequences. (Running on oeis4.)