Search: id:A039834
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%I A039834
%S A039834 1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,
%T A039834 4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,
%U A039834 832040,1346269,2178309,3524578,5702887,9227465,14930352,24157817
%V A039834 1,1,0,1,-1,2,-3,5,-8,13,-21,34,-55,89,-144,233,-377,610,-987,1597,-2584,
%W A039834 4181,-6765,10946,-17711,28657,-46368,75025,-121393,196418,-317811,514229,
%X A039834 -832040,1346269,-2178309,3524578,-5702887,9227465,-14930352,24157817
%N A039834 a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers 
               (A000045) extended to negative indices.
%C A039834 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
%C A039834 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).
%C A039834 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]
%C A039834 The Wikipedia reference was deleted by Wikipedia. [From Cino Hilliard 
               (hillcino368(AT)hotmail.com), Apr 29 2009]
%H A039834 T. D. Noe, <a href="b039834.txt">Table of n, a(n) for n=-2..500</a>
%H A039834 Wikipedia, <a href="http://en.wikipedia.org/wiki/Anti-Fibonacci_number">
               Anti-Fibonacci number</a>.
%F A039834 G.f.: (1+2*x)/(1+x-x^2).
%F A039834 a(n-2)=Sum_{k, 0<=k<=n}(-2)^k*A055830(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 18 2006
%p A039834 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]
%o A039834 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29 
               2009: (Start)
%o A039834 (PARI) /* Simple generation */
%o A039834 fibn(n)=
%o A039834 {
%o A039834 local(a=1,b=1,c);
%o A039834 print1(a","b",");
%o A039834 for(x=3,n,c=a-b;
%o A039834 print1(c",");
%o A039834 a=b;b=c;
%o A039834 );
%o A039834 }
%o A039834 (End)
%Y A039834 Cf. A000045.
%Y A039834 Sequence in context: A107358 A132636 A152163 this_sequence A000045 A020695 
               A132916
%Y A039834 Adjacent sequences: A039831 A039832 A039833 this_sequence A039835 A039836 
               A039837
%K A039834 sign,easy,nice
%O A039834 -2,6
%A A039834 Alexander Grasser (pyropunk(AT)usa.net)
%E A039834 Signs corrected by Len Smiley (smiley(AT)math.uaa.alaska.edu) and N. 
               J. A. Sloane (njas(AT)research.att.com).

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