From: Patsloane-AT-aol.com
Date: Fri, 28 Nov 1997 00:09:07 -0500 (EST)
Subject: PLC: Fibonacci Numbers


OK Metin,

What I want is a formula for finding the nth Fibonacci number, where n can be
any number.  If there is a formula and you know it, just tell me.  Otherwise,
what follows are just observations, not proofs.  I don't know how to prove
any of these things, because what gets me confused is all that combinatorial
stuff about adding and subtracting a series from another series.  

The series of square numbers (1, 4, 9, 16, 25...) alternates an odd number
with an even number.  The series of triangular numbers (1, 3, 6, 10, 15, 21,
28..) alternates 2 odd with 2 even.  The Fibonacci series alternates 1 odd
with 2 even. So this odd-even polarity is interesting.  If there can be a
formula for the nth square number and the nth triangular number, there should
be a formula for the nth Fibonacci number. But I can't figure out what it is.

Thank you.

pat sloane
========================================================Finding the Value of the nth Fibonacci Number

The series of Fibonacci numbers alternates one term which is an 
even number with two terms that are odd numbers. Therefore the nth 
term in the Fibonacci series will fall into one of three classes, according 
to the value of n.

A) Fibonacci numbers that are even numbers. Each of these terms 
is both preceded and followed by a term which is an odd 
number. (n=3x+1) Example: the 1st, 4th, 7th, and 10th 
Fibonacci numbers are even numbers. 

B) Fibonacci numbers that are odd numbers and that directly 
precede a term in the series which is an even number. (n=3x)

C) Fibonacci numbers that are odd numbers and that directly 
follow a term in the series which is an even number. (n=3x+2)

------------------------------------------------------------------------------
---------
 (A) Fibonacci numbers that are even numbers

Let x and n be integers. The nth Fibonacci number is an even 
number if and only if n can be expressed as 3x + 1.

NOTE: The series of Fibonacci number that are even numbers 
alternates terms that are divisible only by 2 with terms that are 
divisible by 8. The nth term in the Fibonacci series is divisible by 8 if 
and only if n can be expressed as 3x + 1, and n is also an odd number. 
Every 6th term in the series that is larger than 0 is divisible by 8.

--------------------------------------------------------------
Generating the series of Fibonacci numbers that are even numbers

Begin with 0 and 2. Each term is 4 times the previous term plus 
the term before that. The series of Fibonacci numbers that are even 
numbers begins, 

0, 2, 8, 34, 144, 610, 2584, 10946...

------------------------------------------------------------------
Generating the series of Fibonacci numbers that are even numbers

The series of Fibonacci numbers that are both even numbers and 
divisible by 8 begins, 

0, 8, 144, 2584, 46368, 832040...

Each term is 18 times the previous term, less the term before 
that. 


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