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A039834 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. 35
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 (list; graph; refs; listen; history; text; internal format)
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

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. - 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

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n = -2..500

M. C. Firengiz, 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.

J. Jina and P. Trojovsky, 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).

Wikipedia, Lucas sequence

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

Index entries for Lucas sequences

FORMULA

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 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)*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

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

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 *)

a[ n_] := Fibonacci[ -n]; (* 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()

[a.next() 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

CROSSREFS

Cf. A000045, A038754, A011782, A215022, A215023.

Sequence in context: A185357 A132636 A152163 * A000045 A236191 A020695

Adjacent sequences:  A039831 A039832 A039833 * A039835 A039836 A039837

KEYWORD

sign,easy,nice

AUTHOR

Alexander Grasser (pyropunk(AT)usa.net)

EXTENSIONS

Signs corrected by Len Smiley and N. J. A. Sloane.

STATUS

approved

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Last modified June 29 10:41 EDT 2016. Contains 274295 sequences.