Definition:Fibonacci Numbers
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Definition
The Fibonacci numbers are a sequence $\left \langle {F_n}\right \rangle$ which is formally defined recursively as:
- $F_0 = 0; \ F_1 = 1; \ F_n = F_{n-1} + F_{n-2}$
That is, the next number in the sequence is found by adding together the two previous ones.
We can see that the first few Fibonacci numbers are:
- $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$
Source of Name
This entry was named for Leonardo Fibonacci.
The name Fibonacci numbers was given to this sequence by Édouard Lucas, who studied these numbers in detail.
The sequence $\left \langle {F_n}\right \rangle$ was known to Indian mathematicians as long ago as the 7th century C.E.
It was also studied by Gopāla before 1135, and by Acharya Hemachandra in about 1150.
Hence some sources refer to these numbers as the Gopala-Hemachandra numbers.
They are also discussed by Johannes Kepler in his work of 1611 De Nive Sexangula (On the Six-Cornered Snowflake).
Also see
This sequence is A000045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
According to the above link, this is also known as Lamé's Sequence, after Gabriel Lamé. However, this suggestion is difficult to corroborate.
- Results about Fibonacci numbers can be found here.
Sources
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968) $\S 1.2.8$
- For a video presentation of the contents of this page, visit the Khan Academy.
