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.
Example: the next number in the sequence above is 21+34 = 55
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, ...
When we make squares with those widths, we 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 Fibonacci Sequence can be written as a "Rule".
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:
Example: term 9 is calculated like this:
x_{9}  = x_{91} + x_{92} 
= x_{8} + x_{7}  
= 21 + 13  
= 34 
And here is a surprise. When we 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 when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we 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 that not just the Fibonacci Sequence can do this!
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.
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:
And so on (every nth number is a multiple of x_{n}).
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  ... 
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.
Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!
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 HinduArabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.
Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!
Source: Math is Fun