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A039834
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a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended to negative indices.
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58
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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) = 5, ..., F(-n) = (-1)^(n-1) F(n). See A215022, A215023 for the negaFibonacci representation of n. - N. J. A. Sloane, Aug 03 2012
The sequence a(n), n >= 0 := 0, 1, -1, 2, -3, 5, -8, 13, ... is the inverse binomial transform of A000045. - Philippe Deléham, 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
This sequence appears in the formula for 1/rho(5)^n, with rho(5) = (1 + sqrt(5))/2 = phi (golden section), when written in the power basis <1, rho(5)> of the quadratic number field Q(rho(5)): 1/rho(5)^n = a(n+1) * 1 + a(n) * rho(5), n >= -2. - Wolfdieter Lang, Nov 04 2013
The sequence 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, ... (starting at offset 1) is the reversion of the g.f. for the "shadows" of Motzkin numbers with offset 1 (see A343773). - Gennady Eremin, Jul 16 2021
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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).
D. Shtefan and I. Dobrovolska, The sums of the consecutive Fibonacci numbers, Fib. Q., 56 (2018), 229-236.
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LINKS
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FORMULA
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G.f.: (1+2*x)/(x^2*(1+x-x^2)).
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)*Fibonacci(k)*(-1)^(n-k), n > 0, 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
G.f.: 1/(G(0)*x^3) + (2*x^2+x-1)/x^3, where G(k) = 1 + 2*x*(k+1)/(k + 2 - x*(k+2)*(k+3)/(x*(k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 27 2013
G.f.: Q(0)/x - 1/x + 1+ x, where Q(k) = 1 + x^2 + x^3 + k*x*(1+x^2) - x^2*(1 + x*(k+2))*(1+k*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 13 2014
a(n) = -(-1)^n*A000045(n), at least for all n >= 0 (and also for n < 0 if A000045 is extended to negative indices). - M. F. Hasler, May 10 2017
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EXAMPLE
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With the golden section phi = rho(5) = (1 + sqrt(5))/2:
n = -2: phi^2 = a(-1)*1 + a(-2)*phi = 1 + phi,
n = -1: phi = a(0)*1 + a(-1)*phi = phi, (trivial)
n = 0: 1/phi^0 = a(1)*1 + a(0)*phi = 1, (trivial)
n = 1: 1/phi = a(2)*1 + a(1)*phi = -1 + phi,
n = 2: 1/phi^2 = a(3)*1 + a(2)*phi = 2 - phi, ... (End)
G.f. = x^-2 + x^-1 + x - x^2 + 2*x^3 - 3*x^4 + 5*x^5 - 8*x^6 + 13*x^7 - ...
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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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PROG
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(PARI) a(n) = fibonacci(-n);
(Haskell)
a039834 n = a039834_list !! (n+2)
a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list)
(Sage)
x, y = 1, 1
while True:
yield x
x, y = y, x - y
(Sage)
R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len)
f = (-2*t-1)/(t^4-t^3-t^2)
return f.list()
(Python)
from sympy import fibonacci
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CROSSREFS
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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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STATUS
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approved
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