-2,6

Starting with (a(-1), a(0), a(1), a(2)) = (1, 0, 1, -1) gives the subsequence called the "anti-Fibonacci numbers" [see Wikipedia]. The ratio of successive anti-Fibonacci numbers converges to -1/phi. - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 10 2006

Comment from Bill Gosper, May 28 2008: 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).

The sequence a(n), n>=0 := 0,1,-1,2,-3,5,-8,13,... is the inverse binomial transform of A000045. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

The Wikipedia reference was deleted by Wikipedia. [From Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29 2009]

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

Wikipedia, Anti-Fibonacci number.

G.f.: (1+2*x)/(1+x-x^2).

a(n-2)=Sum_{k, 0<=k<=n}(-2)^k*A055830(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2006

a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1, 2]: seq (a(n), n=-2..50); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Nov 01 2008]

Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29 2009: (Start)

(PARI) /* Simple generation */

fibn(n)=

{

local(a=1, b=1, c);

print1(a", "b", ");

for(x=3, n, c=a-b;

print1(c", ");

a=b; b=c;

);

}

(End)

Cf. A000045.

Sequence in context: A107358 A132636 A152163 this_sequence A000045 A020695 A132916

Adjacent sequences: A039831 A039832 A039833 this_sequence A039835 A039836 A039837

sign,easy,nice

Alexander Grasser (pyropunk(AT)usa.net)

Signs corrected by Len Smiley (smiley(AT)math.uaa.alaska.edu) and N. J. A. Sloane (njas(AT)research.att.com).