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A039834
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a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices.
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28
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1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233, -377, 610, -987, 1597, -2584, 4181, -6765, 10946, -17711, 28657, -46368, 75025, -121393, 196418, -317811, 514229, -832040, 1346269, -2178309, 3524578, -5702887, 9227465, -14930352, 24157817
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OFFSET
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-2,6
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COMMENTS
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Knuth defines the negaFibonacci numbers as follows: F_{-1}=1, F_{-2}=-1, F_{-3}=2, F_{-4}=-3, F_{-5}=8, ..., F_{-n}=(-1)^(n-1)F_n. See A215022, A215023 for the negaFibonacci representation of n. - N. J. A. Sloane, Aug 03 2012
The ratio of successive terms converges to -1/phi. - Jonathan Vos Post, Dec 10 2006
Let a[n]:=fib[n]*(-1)^binom(n,2). Then a[m-n]*a[m+n] = a[m+1]*a[m-1]*a[n]^2 - a[n+1]*a[n-1]*a[m]^2. This plus gcd(f[n],f[m]) = |f[gcd(n,m)]| makes a[] a strong elliptic divisibility sequence. Likewise fib[n]*(-1)^binom(n-1,2), but no other asSIGNation (mod scaling). [Bill Gosper, May 28 2008]
The sequence a(n), n>=0 := 0,1,-1,2,-3,5,-8,13,... is the inverse binomial transform of A000045. [From Philippe DELEHAM, Oct 28 2008]
Equals the INVERTi transform of A038754, assuming that an additional A038754(0)=1 is added in front of A038754, and that the a(n) are prefixed with another 1 and then get offset 0. - Gary W. Adamson, Jan 08 2011
If we remove a(-2) and then set the offset to 0, we have the INVERT transform of a signed A011782: (1, -1, 2, -4, 8, -16, 32,...).- Gary W. Adamson, Jan 08 2011
The sequence 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144,.. (starting at offset 0) is the Lucas U(-1,-1) sequence. - R. J. Mathar, Jan 08 2013
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 168, Eq. (145). - N. J. A. Sloane, Aug 03 2012
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LINKS
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T. D. Noe, Table of n, a(n) for n=-2..500
Wikipedia, Lucas sequence
Index to sequences with linear recurrences with constant coefficients, signature (-1,1)
Index entries for Lucas sequences
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FORMULA
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G.f.: (1+2*x)/(x^2*(1+x-x^2)).
a(n-2) = Sum_{k, 0<=k<=n}(-2)^k*A055830(n,k). - Philippe DELEHAM, Oct 18 2006
a(n) = ((phi - 1)^n + 1/phi*(-(1/phi) - 1)^(n+1))/sqrt(5), where phi = (1 + sqrt(5))/2. - Arkadiusz Wesolowski, Oct 28 2012
a(n) = sum(k=1..n, binomial(n-1,k-1)*fib(k)*(-1)^(n-k)), n>0, fib(k)=A000045(k), a(0)=1. [Perminova Maria, Jan 22 2013]
G.f.: 1 + x/( Q(0) - x ) where Q(k) = 1 - x/(x*k - 1 )/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
G.f.: 2 - 2/(Q(0)+1) where Q(k) = 1 + 2*x/(1 - x/(x + 1/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
G.f.: 1+x^2+x^3 + x/Q(0) , where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013
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MAPLE
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a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1, 2]: seq (a(n), n=-2..50); # Alois P. Heinz, Nov 01 2008
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MATHEMATICA
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LinearRecurrence[{-1, 1}, {1, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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PROG
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(PARI) a(n) = fibonacci(-n);
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CROSSREFS
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Cf. A000045, A038754, A011782, A215022, A215023.
Sequence in context: A185357 A132636 A152163 * A000045 A020695 A212804
Adjacent sequences: A039831 A039832 A039833 * A039835 A039836 A039837
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KEYWORD
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sign,easy,nice
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AUTHOR
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Alexander Grasser (pyropunk(AT)usa.net)
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EXTENSIONS
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Signs corrected by Len Smiley (smiley(AT)math.uaa.alaska.edu) and N. J. A. Sloane.
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STATUS
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approved
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