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A099496
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(-1)^n*Fib(2n+1).
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3
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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; internal format)
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OFFSET
| 0,2
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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)).
Contribution from Adam Helman (helman(AT)san.rr.com), May 09 2010: (Start)
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.
(End)
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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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)], with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Apr 13 2007
a(n)=-3*a(n-1)-a(n-2),a(0)=1, a(1)=-2. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]
Contribution from Adam Helman (helman(AT)san.rr.com), May 09 2010: (Start)
a(n) = ceil( 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). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2010]
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EXAMPLE
| Contribution from Adam Helman (helman(AT)san.rr.com), May 09 2010: (Start)
The first term: a(1) = ceil( phi / [F_2/F_1 - phi] ) = -2. (End)
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MATHEMATICA
| lst={}; Do[AppendTo[lst, (-1)^n*Fibonacci[2*n+1]], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 18 2009]
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CROSSREFS
| Sequence in context: A001519 A122367 A048575 * A114299 A112842 A097417
Adjacent sequences: A099493 A099494 A099495 * A099497 A099498 A099499
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Oct 19 2004
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