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2012-11-24, 01:02   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

22·3·79 Posts
Fibonacci Formula

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

Regards,
Matt
Attached Files
 Another Fibonacci Equation.pdf (226.1 KB, 188 views)

 2012-11-24, 01:18 #2 Dubslow Basketry That Evening!     "Bunslow the Bold" Jun 2011 40
2012-11-24, 03:23   #3
LaurV
Romulan Interpreter

"name field"
Jun 2011
Thailand

24·613 Posts

Quote:
 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...

 2012-11-24, 10:17 #4 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 22×3×79 Posts 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
 2012-11-24, 23:28 #5 Stargate38     "Daniel Jackson" May 2011 14285714285714285714 677 Posts Exact formula. Here's a formula for the Fibonacci numbers. It even works for negative indices: Fn=(phin-(-phi)-n)/sqrt(5), where phi is the Golden Ratio: (sqrt(5)+1)/2.
2013-01-08, 20:05   #6
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

22·3·79 Posts

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.
Attached Files
 Fibonacci Equation.pdf (156.5 KB, 171 views)

 2013-01-11, 13:12 #7 Nick     Dec 2012 The Netherlands 110110111102 Posts 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.
 2013-01-14, 23:29 #8 Stargate38     "Daniel Jackson" May 2011 14285714285714285714 12458 Posts There is also a formula that uses the Lucas numbers: Fn*Ln=F2n :)

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