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 ratio. 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, beginning with
8/5 = 1.60000000000...
13/8 = 1.6250000000...
21/13 = 1.615384615...
34/21 = 1.619047619...
55/34 = 1.617647059...
89/55 = 1.618181818...
you can see that they come closer and closer to the golden ratio, which is the quadratic irrational number
(1 + √5)/2 = 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137...
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 founded 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 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
A remarkable resource for studying Fibonacci sequences and related sequences is the Online Encyclopedia of Integer Sequences: OEIS.
On the first day of 2025, OEIS included more than 378,000 sequences. The following table shows the number of hits in OEIS for various searches:
Fibonacci: 10847 hits
Lucas: 3188 hits
Hoggatt: 216 hits
Brousseau: 70 hits
Zeckendorf: 493 hits
Wythoff: 583 hits
Beatty: 727 hits
linear recurrence: 39411 hits
infinite Fibonacci word: 243 hits
Here are a few specific sequences in OEIS that can be downloaded with a click:
A000045: Fibonacci sequence
A000032: Lucas sequence
A003714: Fibbinary numbers
A007895: Number of terms in Zeckendorf representations
A035517: Zeckendorf representations
A000201: Lower Wythoff sequence
A001950: Upper Wythoff sequence
A035513: Wythoff array
A003849: Infinite Fibonacci word (010010...)
A014675: Infinite Fibonacci word (212212...)
A001622: Golden ratio
A180662: Golden triangle (products of consecutive Fibonacci numbers
A152149: Doubly golden triangle (unique geometric shape)
A376961: Doubly golden triangle (sidelengths)
A226080: Rabbit tree
The founder of OEIS, Neil J. A. Sloane, was the keynote speaker at the Fifteenth Conference on Fibonacci Numbers and Their Applicatons, Eger, Hungary, June 2012. A paper based on his talk can be downloaded here: Title of talk: The On-Line Encyclopedia of Integer Sequences. See also Homepage of the Fifteenth International Conference on Fibonacci Numbers and Their Applications.
Following are group pictures of Fibonacci number-theorists and other participants in international conferences of the Fibonacci Association.
Left-to-right: Dan Fielder, Norbert Jensen, Calvin T. Long, Marjorie Johnson, Herta Hoggatt, Jerry Bergum.
Photo by Frank Johnson.
For access to many other Fibonacci-related resources, visit The Fibonacci Association.