1. Fibonacci Sequence and Related Numbers
The famous Fibonacci sequence is the result of a question posed by Leonardo de Fibonacci, a mathematician during the Middle Ages.
If you begin with one pair of rabbits on the first day of the year, how many pairs of rabbits will you have on the first day of the next year?
It is assumed that each pair of rabbits produces a new pair every month and each new pair begins to produce two months after birth.

2. The Fibonacci Problem
How Many Pairs of Rabbits Are Created by One Pair in One Year
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

3. Fibonacci cont’l
He died in the 1240's and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa.

4. Fig. 2-69, p. 113

5. Fig. 2-70, p. 114

6. Fibonacci Sequence, cont’d
The solution to this question is shown in the table below.
The sequence that appears three times in the table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the Fibonacci sequence.

7. Fibonacci Sequence, cont’d
The Fibonacci sequence is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …
The Fibonacci sequence is found in many places in nature.
Any number in the sequence is called a Fibonacci number.
The sequence is usually written F1, F2, F3, …, Fn, …

8. Recursion
Recursion, in a sequence, indicates that each number in the sequence is found using previous numbers in the sequence.
Some sequences, such as the Fibonacci sequence, are generated by a recursion rule along with starting values for the first two, or more, numbers in the sequence.

9. Fibonacci Numbers In Nature
The Fibonacci numbers are found many places in the natural world, including:
The number of flower petals.
The branching behavior of plants.
The growth patterns of sunflowers and pinecones, ……
It is believed that the spiral nature of plant growth accounts for this phenomenon.

10. Fibonacci Numbers In Nature, cont’d
The number of petals on a flower are often Fibonacci numbers.

13. Fibonacci Numbers In Nature, cont’d
Plants grow in a spiral pattern. The ratio of the number of spirals to the number of branches is called the phyllotactic ratio.
The numbers in the phyllotactic ratio are usually Fibonacci numbers.

14. Fibonacci Numbers In Nature, cont’d
Mature sunflowers have one set of spirals going clockwise and another set going counterclockwise.
The numbers of spirals in each set are usually a pair of adjacent Fibonacci numbers.
The most common number of spirals is 34 and 55.