Fibonacci Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it.
 The 2 is found by adding the two numbers before it (1+1)
 Similarly, the 3 is found by adding the two numbers before it (1+2),
 And the 5 is (2+3),
 and so on!
Example: the next number in the sequence above would be 21+34 = 55
It is that simple!
Here is a longer list:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...
Can you figure out the next few numbers?
Makes A Spiral
When you make squares with those widths, you get a nice spiral:
Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.
The Rule
The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series).
First, the terms are numbered from 0 onwards like this:
n =  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  ... 
x_{n} =  0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  ... 
So term number 6 is called x_{6} (which equals 8).
Example: the 8th term is

So we can write the rule:
The Rule is x_{n} = x_{n1} + x_{n2}
where:
 x_{n} is term number "n"
 x_{n1} is the previous term (n1)
 x_{n2} is the term before that (n2)
Example: term 9 would be calculated like this:
x_{9} = x_{91} + x_{92} = x_{8} + x_{7} = 21 + 13 = 34
Golden Ratio
And here is a surprise. If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034... In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few: 
A 
B 
B / A 


2 
3 
1.5  
3 
5 
1.666666666...  
5 
8 
1.6  
8 
13 
1.625  
... 
... 
...  
144 
233 
1.618055556...  
233 
377 
1.618025751...  
... 
... 
... 
Note: this also works if you pick two random whole numbers to begin the sequence, such as 192 and 16 (you would get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):
A 
B 
B / A 


192 
16 
0.08333333...  
16 
208 
13  
208 
224 
1.07692308...  
224 
432 
1.92857143...  
... 
... 
...  
7408 
11984 
1.61771058...  
11984 
19392 
1.61815754...  
... 
... 
... 
It takes longer to get good values, but it shows you that it is not just the Fibonacci Sequence that can do this!
Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:
The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.
Example:
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.
Try it for yourself!
A Pattern
Here is the Fibonacci sequence again:
n =  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  ... 
x_{n} =  0  1  1  2  3  5  8  13  21  34  55  89  144  233  377  610  ... 
There is an interesting pattern:
 Look at the number x_{3} = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)
 Look at the number x_{4} = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)
 Look at the number x_{5} = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)
And so on (every nth number is a multiple of x_{n}).
Terms Below Zero
The sequence works below zero also, like this:
n =  ...  6  5  4  3  2  1  0  1  2  3  4  5  6  ... 
x_{n} =  ...  8  5  3  2  1  1  0  1  1  2  3  5  8  ... 
(Prove to yourself that each number is found by adding up the two numbers before it!)
In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a ++ ... pattern. It can be written like this:
x_{−n} = (−1)^{n+1} x_{n}
Which says that term "n" is equal to (−1)^{n+1}^{ } times term "n", and the value (−1)^{n+1} neatly makes the correct 1,1,1,1,... pattern.
History
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!
About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread through Europe the use of HinduArabic Numerals (like our present number system 0,1,2,3,4,5,6,7,8,9) to replace Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.