Dear Enuf:

Well, it's the famous Fibonacci sequence. One reason it's famous is that appears in so many often unexpected place and has so many different and curious properties that numerous chapters have been written about it. Named after Leonardo of Pisa son of Bonacci who described it in 1202 in his "Liber abaci".

The sequence is intimately related to the golden ratio t (I'll write t instead of ususal tau.) t = (1+5)/2.

If f_n is the n-th Fibonacci number so that f₀=0, f₁=1, f₂=1, f₃=2, f₄=3, ... and thus f_n+1 = f_n + f_n-1, then

f_n = (tⁿ - (1-t)ⁿ)/5 which is due to Euler (1765).

The formula itself is remarkable. Writing it explicitly you may wonder how such an apparently irrational expression may lead to an integer number.

I'll list a few properties of the sequence:

1. The ratio f_n+1/f_n approaches the golden ratio as n grows.
2. f_n+1*f_n-1 - f_n² = (-1)ⁿ (Cassini's formula (1680))
3. f_n and f_n+1 are always relatively prime.
4. Any integer is a sum of distinct Fibonacci numbers.
5. f_kn is a multiple of f_n. Moreover,
6. gcd(f_m, f_n) = f_{gcd(m, n)}, (gcd - greatest common divisor)

You may be curious to look into the following books (check my Book store for details):

1. M.Gardner, Mathematical Circus
2. R.L.Graham et al, Concrete Mathematics
3. H.S.M.Coxeter, Introduction to Geometry

On Internet look into:

http://mathforum.org/dr.math/tocs/golden.high.html

or just run a search on the "Fibonacci Sequence" - results are sure to be overwhelming.

Best regards,
Alexander Bogomolny

Copyright © 1996-2009 Alexander Bogomolny

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