OFFSET
-2,6
COMMENTS
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 ratio of successive terms converges to -1/phi. - Jonathan Vos Post, Dec 10 2006
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
a(n) = A227431(n + 4, n + 3). - Reinhard Zumkeller, Feb 01 2014
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
REFERENCES
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.
LINKS
Indranil Ghosh, Table of n, a(n) for n = -2..4773 (terms -2..500 from T. D. Noe)
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.
M. Cetin Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
Jiřı Jina and Pavel Trojovský, On determinants of some tridiagonal matrices connected with Fibonacci numbers, International Journal of Pure and Applied Mathematics, Volume 88 No. 4 2013, 569-575; ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version).
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2.
J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
Emil Daniel Schwab and Gabriela Schwab, k-Fibonacci numbers and Möbius Functions, Integers (2022) Vol. 22, #A64.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (-1,1).
FORMULA
G.f.: (1+2*x)/(x^2*(1+x-x^2)).
a(n-2) = Sum_{k=0..n} (-2)^k*A055830(n, k). - Philippe Deléham, 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)*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
a(n) = Sum_{k=0..floor((n-1)/2)} A130595(n-k-1, k), for n >= 0. - G. C. Greubel, Jun 22 2024
EXAMPLE
From Wolfdieter Lang, Nov 04 2013: (Start)
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 - ...
MAPLE
a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1, 2]: seq(a(n), n=-2..50); # Alois P. Heinz, Nov 01 2008
MATHEMATICA
LinearRecurrence[{-1, 1}, {1, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
Fibonacci[-Range[-2, 37]] (* Michael Somos, Jun 04 2016 *)
PROG
(PARI) a(n) = fibonacci(-n);
(Haskell)
a039834 n = a039834_list !! (n+2)
a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list)
-- Reinhard Zumkeller, Jul 05 2013
(Sage)
def A039834():
x, y = 1, 1
while True:
yield x
x, y = y, x - y
a = A039834()
[next(a) for i in range(40)] # Peter Luschny, Jul 11 2013
(Sage)
def A039834_list(len):
R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len)
f = (-2*t-1)/(t^4-t^3-t^2)
return f.list()
A039834_list(40) # Peter Luschny, Nov 21 2014
(Magma) [Fibonacci(-n): n in [-2..40]]; // Marius A. Burtea, Nov 14 2019
(Python)
from sympy import fibonacci
def A039834(n): return fibonacci(-n) # Chai Wah Wu, Jan 20 2022
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
Alexander Grasser (pyropunk(AT)usa.net)
EXTENSIONS
Signs corrected by Len Smiley and N. J. A. Sloane
STATUS
approved