Math Explorers Club
We’ve already seen how the Fibonacci numbers work, and gotten a good idea of how to get a lot of Fibonacci equalities from combinatorially representations. However, there are lots of numbers that act “kind of” like the Fibonacci numbers.
What do we mean by this? First, we had the relation for the Fibonacci numbers given by
along with the initial conditions 
and 
. However, any numbers that satisfy the Fibonacci relation above - but with different initial conditions - can still have nice properties like the Fibonacci numbers.
As an example, try the following exercise.
Exercise 1. Let a,b be positive integers. Form the sequence defined by 
, 
and 
for all 
. (Remember, if 
, these will be the Fibonacci numbers.)
Show that as 
goes to infinity, the ratio of 
to 
approaches the golden ratio, i.e.
Note that this worked for any positive integers a and b. Sequences of this form will be called “gibonacci sequences” - as in “generalized Fibonacci sequences.” A particularly famous (and common) one of these sequences is called the Lucas numbers, obtained when we take a=2 and b=1.
The first few terms of this sequence are 2,1,3,4,7, and 11.
Lesson 4: Lucas Numbers
Tuesday, July 1, 2008
Section 1. Almost Fibonacci?
