Fibonacci Numbers

1. Fibonacci Numbers

2. Fibonacci Numbers • In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci, whose Liber Abaci published in 1202 introduced the sequence to Western European mathematics. • The sequence is defined by the following recurrence relation: • That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as Fn, for n = 0, 1, … , are: 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, … • Each third number of the series is an even number. • The sequence named after Fibonacci was first described in Indian mathematics. • The sequence extended to negative index n satisfies Fn = Fn−1 + Fn−2 for all integers n, and F-n = (−1)n+1Fn: • .., -8, 5, -3, 2, -1, 1, followed by the sequence above.

3. Fibonacci Rectangles

4. 1

5. 1 1

6. 1 1 2

7. 1 1 3 2

8. 1 1 3 2 5

9. 8 1 1 3 2 5

10. 13 8 1 1 3 2 5

11. 13 21 8 1 1 3 2 5