Difference between revisions 30624594 and 39413900 on enwiki

The '''anti-Fibonacci numbers''' compose a sequence closely related to the [[Fibonacci sequence]].  It satisfies the following [[recursion]] formula:

:<math>f(k+2) = f(k) - f(k+1)\,</math>

Usually, it is made to start with 1, 0. Then, it goes like this:

:<math>1,0,1,-1,2,-3,5,-8, \dots\,</math>

For easy comparison: The Fibonacci sequence satisfies

:<math>f(k+2) = f(k) + f(k+1)\,</math>

and goes like this:

:<math>0,1,1,2,3,5,8, \dots\,</math>
 
If the Fibonacci sequence is extended backwards -

:<math>\dots -8,5,-3,2,-1,1,0,1,1,2,3,5,8, \dots\,</math>

- the anti-Fibonacci numbers also appear.⏎
chicken chicken

While the ratio <math>\frac{f(k+1)}{f(k)}</math> converges to <math>\phi=1.618</math> for the Fibonacci sequence, the ratio converges to <math>-\frac{1}{\phi}=-0.618</math> for the anti-Fibonacci numbers.

[[Category:Integer sequences]]