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.
(forthcoming)
For access to many other Fibonacci-related resources, visit The Fibonacci Association.