Fibonacci Formula

[MattcAnderson]

Hi all,

I thought I would tell about a little known Fibonacci formula that I found, and then later found it in the literature. A deffinition of Fibonacci numbers is -
F(n) = F(n-1) + F(n-2) with F(0) = F(1) = 1
This is a formula that lets you compute F(2n), knowing F(n) and F(n-1).
The most general form is -

F(n) = F(k)*F(n-k) + F(k-1)*F(n-k-1)

substituting n=2k or n=2k+1 gives interesting results.
I will try to link to my derivation

https://docs.google.com/viewer?a=v&p...AwMGIzOTU5NjRk

Regards,
Matt

[Dubslow]

Take a look at the bottom of this section, and this section. The fourth formula of the latter seems to contradict one of your special cases, and it also has a general formula for F_kn+c.

Though your formula doesn't appear, I'd guess that someone's probably thought of it before given it's rather simple derivation.

[LaurV]

Originally Posted by MattcAnderson

F(0) = F(1) = 1

I believe that you "destroy" a lot of mathematical (and also biological ) things with this unintentional shift...
For example you can not say anymore that "the only fibo numbers that can be prime are those with a prime index".... and many other such affirmations...

[MattcAnderson]

Hi,

Thanks for the replies. You are right, the standard definition of Fibonacci numbers includes F(1) = F(2) = 1. Mentioning that F(0) = 1 was an error.

With this change, the formulas currently at the bottom of the Wikipedia article sections cited are exactly the same ones I mentioned.

-Matt

[Stargate38]

Here's a formula for the Fibonacci numbers. It even works for negative indices:

F_n=(phiⁿ-(-phi)^-n)/sqrt(5),

where phi is the Golden Ratio: (sqrt(5)+1)/2.

[MattcAnderson]

I corrected my derivation. This formula is useful because a person can calculate large Fibonacci numbers F(n) with integer arithmatic and fewer steps than the definition equation. For example, if someone wants to calculate F(100), Start with F(1), F(2), and F(3) then find F(6), F(12), F(24), F(25), F(50), and finally F(100). This is shorter than iterating 100 times.

[Nick]

Hendrik Lenstra wrote a magazine article about profinite Fibonacci numbers:
http://www.math.leidenuniv.nl/~hwl/papers/fibo.pdf

With profinite integers, you get 8 extra fixed points.

[Stargate38]

There is also a formula that uses the Lucas numbers:

F_n*L_n=F_2n :)