For centuries, mathematicians—both amateurs and professionals—have been intrigued by the sequence of Fibonacci numbers and the closely related irrational number called the
golden mean. The sequence begins with
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . ,where, as you can see, each number beginning with 2 is the sum of the two immediately preceding numbers.
As you progress along the list of quotients of consecutive Fibonacci numbers, such as
8/5, 13/8, 21/13, 34/21, . . . ,
you get closer and closer to the golden mean, which is exactly (1 + SQR(5))/2 and approximately 1.618033989.
The Fibonacci sequence belongs to an extensive subject called recurrence sequences. In 1963, a group of number-theorists under the leadership of Vern Hoggatt and Brother Alfred Brousseau formed the Fibonacci Association, and they started a still thriving journal, The Fibonacci Quarterly. The journal specializes in recurrence sequences and their applications. Among the names most often encountered in pages of the Quarterly are these:
Fibonacci (c.1175-c.1240) as in Fibonacci numbers
Édouard Lucas (1842-1891) as in Lucas numbers
Verner Emil Hoggatt, Jr. (1921-1981) co-founder of the Fibonacci Association
Brother Alfred Brousseau, F.S.C. (1907-1988) co-founder of the Fibonacci Association
Edouard Zeckendorf (1901-1983) as in Zeckendorf sums
Willem Abraham Wythoff (1865-1939) as in Wythoff game
Samuel Beatty (1881-1970) as in Beatty sequences