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A099496 a(n) = (-1)^n * Fibonacci(2n+1). 4
1, -2, 5, -13, 34, -89, 233, -610, 1597, -4181, 10946, -28657, 75025, -196418, 514229, -1346269, 3524578, -9227465, 24157817, -63245986, 165580141, -433494437, 1134903170, -2971215073, 7778742049, -20365011074, 53316291173, -139583862445, 365435296162, -956722026041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)).

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

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..2386

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (-3,-1).

FORMULA

G.f.: (1+x)/(1+3x+x^2);(with interpolated zeros) (1+x^2)/(1+3x^2+x^4); a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*cos((n-2k)*pi/2)} (with interpolated zeros); a(n)=F(n+1)(-1)^(n/2)(1+(-1)^n)/2 (with interpolated zeros).

a(n) = (-1)^n * Sum_{k=0..n+1} binomial(n+k,n-k). - Paolo P. Lava, Apr 13 2007

a(n) = -3*a(n-1)-a(n-2),a(0)=1, a(1)=-2. - Philippe Deléham, Nov 03 2008

From Adam Helman (helman(AT)san.rr.com), May 09 2010: (Start)

a(n) = ceiling( phi / [F_{n+1}/F_n - phi] ).

An exact expression for the inverse fractional error is phi / [F_{n+1}/F_n - phi] = (phi/sqrt(5)) * [(-1)^n *{phi^2n} - 1]. (End)

a(n) = (-1)^n*A122367(n). - R. J. Mathar, Jul 23 2010

EXAMPLE

The first term: a(1) = ceil( phi / [F_2/F_1 - phi] ) = -2. - Adam Helman (helman(AT)san.rr.com), May 09 2010

a(3) = (-1)^3 * Fibonacci(2 * 3 + 1) = -Fibonacci(7) = -13. - Indranil Ghosh, Feb 04 2017

MAPLE

seq((-1)^n*combinat:-fibonacci(2*n+1), n=0 .. 100); # Robert Israel, Jul 02 2015

MATHEMATICA

lst={}; Do[AppendTo[lst, (-1)^n*Fibonacci[2*n+1]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 18 2009 *)

Table[(-1)^n Fibonacci[2n+1], {n, 0, 30}] (* Harvey P. Dale, Aug 22 2016 *)

PROG

(MAGMA) [(-1)^n*Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 04 2015

CROSSREFS

Cf. A056594, A122367.

Sequence in context: A001519 A048575 * A122367 A114299 A112842 A097417

Adjacent sequences:  A099493 A099494 A099495 * A099497 A099498 A099499

KEYWORD

easy,sign

AUTHOR

Paul Barry, Oct 19 2004

STATUS

approved

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Last modified November 23 09:38 EST 2017. Contains 295115 sequences.