Difference between revisions 81567909 and 81568127 on enwiki

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

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

The anti-Fibonacci sequence begins

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

By comparison, the Fibonacci sequence satisfies the formula

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

and begins

:<math>0,1,1,2,3,5,8, \dots\,</math>
 
The anti-Fibonacci sequence also appears if Fibonacci sequence is extended backwards (below 0):

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

While the ratio of successive Fibonacci numbers converges on <math>{\phi}\, </math> (the [[golden ratio]]), the ratio of successive anti-Fibonacci numbers converges on <math>-\frac{1}{\phi\,}.</math>

[[Category:Recurrence relations]]
[[Category:Integer sequences]]
[[Category:Fibonacci numbers]]

[[eo:Vikipedio:Projekto matematiko/Kontraŭ-fibonaĉi-aj nombroj]]