Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

FIBONACCI NUMBERS

It may be hard to define mathematical beauty,

but that is true of beauty of any kind.

— G.H. Hardy (1877–1947), A Mathematician's Apology

2.1 FIBONACCI'S RABBITS

Fibonacci's Classic work, Liber Abaci, contains many elementary problems, including the following famous problem about rabbits:

Suppose there are two newborn rabbits, one male and the other female. Find the number of rabbits produced in a year if:

• Each pair takes one month to become mature;
• Each pair produces a mixed pair every month, beginning with the second month; and
• Rabbits are immortal.

Suppose, for convenience, that the original pair of rabbits was born on January 1. They take a month to become mature, so there is still only one pair on February 1. On March 1, they are two months old and produce a new mixed pair, a total of two pairs. Continuing like this, there will be three pairs on April 1, five pairs on May 1, and so on; see the last row of Table 2.1.

Table 2.1 Growth of the Rabbit Population

Number of Pairs	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug
Adults	0	1	1	2	3	5	8	13
Babies	1	0	1	1	2	3	5	8
Total	1	1	2	3	5	8	13	21

2.2 FIBONACCI NUMBERS

The numbers in the bottom row are called Fibonacci numbers, and the sequence is the Fibonacci sequence. Table A.2 in the Appendix lists ...