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%I #69 Feb 23 2024 07:32:36
%S 1,-2,5,-13,34,-89,233,-610,1597,-4181,10946,-28657,75025,-196418,
%T 514229,-1346269,3524578,-9227465,24157817,-63245986,165580141,
%U -433494437,1134903170,-2971215073,7778742049,-20365011074,53316291173,-139583862445,365435296162,-956722026041
%N a(n) = (-1)^n * Fibonacci(2*n+1).
%C With interpolated zeros, a Chebyshev transform of A056594, which has g.f. 1/(1+x^2). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).
%C a(n) is the ceiling of the inverse fractional error in approximating phi, the golden section, by the ratio of two successive terms in the Fibonacci series. - Adam Helman (helman(AT)san.rr.com), May 09 2010
%H Indranil Ghosh, <a href="/A099496/b099496.txt">Table of n, a(n) for n = 0..2386</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-1).
%F G.f.: (1+x)/(1+3x+x^2).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*cos((n-2k)*Pi/2) (with interpolated zeros);
%F a(n) = Fibonacci(n+1)(-1)^(n/2)(1 + (-1)^n)/2 (with interpolated zeros).
%F a(n) = -3*a(n-1) - a(n-2), a(0)=1, a(1)=-2. - _Philippe Deléham_, Nov 03 2008
%F From Adam Helman (helman(AT)san.rr.com), May 09 2010: (Start)
%F a(n) = ceiling( phi / (Fibonacci(n+1)/Fibonacci(n) - phi) ).
%F An exact expression for the inverse fractional error is phi / (Fibonacci(n+1)/Fibonacci(n) - phi) = (phi/sqrt(5)) * ((-1)^n *(phi^2n) - 1). (End)
%F a(n) = (-1)^n*A122367(n). - _R. J. Mathar_, Jul 23 2010
%e a(3) = (-1)^3 * Fibonacci(2 * 3 + 1) = -Fibonacci(7) = -13. - _Indranil Ghosh_, Feb 04 2017
%p seq((-1)^n*combinat:-fibonacci(2*n+1), n=0 .. 100); # _Robert Israel_, Jul 02 2015
%t Table[(-1)^n Fibonacci[2 n + 1], {n, 0, 30}] (* _Harvey P. Dale_, Aug 22 2016 *)
%o (Magma) [(-1)^n*Fibonacci(2*n+1): n in [0..40]]; // _Vincenzo Librandi_, Jul 04 2015
%Y Cf. A056594, A122367.
%K easy,sign
%O 0,2
%A _Paul Barry_, Oct 19 2004
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