|
| |
|
|
A039834
|
|
a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices.
|
|
24
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| -2,6
|
|
|
COMMENTS
| 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, 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, Oct 28 2008]
The Wikipedia reference was deleted by Wikipedia. [From Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29 2009]
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
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=-2..500
Wikipedia, Anti-Fibonacci number (archived in Jan 2007).
Index to sequences with 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<=k<=n}(-2)^k*A055830(n,k). - Philippe DELEHAM, Oct 18 2006
|
|
|
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] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
|
|
|
PROG
| 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)
|
|
|
CROSSREFS
| Cf. A000045, A038754, A011782
Sequence in context: A185357 A132636 A152163 * A000045 A020695 A132916
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 (smiley(AT)math.uaa.alaska.edu) and N. J. A. Sloane (njas(AT)research.att.com).
Deleted Wikipedia link replaced with archive.org's version by Robert Munafo (mrob27(AT)gmail.com), Dec 16 2009
|
| |
|
|
