If you have read Dan Brown's book The Da Vinci Code (or seen the film) you will recognise the scene when Sauniere includes this number sequence as one of the clues he passes on to his grandchild and Robert Langdon.
The Fibonacci number sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34. Each number is the sum of the two numbers that precede it. This can be implemented in betting as you only need two wins in a row to come out on top. You start by betting the first number on the series, i.e. 1. If you lose you move to the number immediately to the right. If you win, you go to the number to the left. When you have won twice in a row the Fibonacci system starts from the beginning.
1. If you bet $1, lose.
2. If you bet on the number to the right, i.e. $1 and lose
3. If you bet $2 (next number), lose
4. If you bet $3, lose
5. If you bet $5, win
6. If you bet $8 win
As you have won twice in a row the system starts over again and the next bet is $1 again. In this example your pay-out is 8+5-3-2-1-1 = $6.
This system is not for the faint-hearted because it has you betting quite high amounts after a few losses in a row.