The Fibonacci sequence is an infinite integer sequence in which each term after the first two is the sum of the two preceding terms. Formally, it is defined by the recurrence relation:
- $F_0 = 0$
- $F_1 = 1$
- $F_n = F_{n-1} + F_{n-2}$ for $n \ge 2$
The resulting series begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ….
History
The sequence was introduced to Western mathematics by Leonardo of Pisa (c. 1170–c. 1250), known as Fibonacci, in his 1202 book Liber Abaci. Although Fibonacci used the sequence to model rabbit population growth, earlier Indian mathematicians, notably in the works of Pingala and later in the Ganita Sara Samgraha (c. 600 CE), described related sequences.
Mathematical Properties
- Closed‑form (Binet's Formula):
$F_n = \frac{\varphi^{,n} - (1-\varphi)^{,n}}{\sqrt{5}}$, where $\varphi = \frac{1+\sqrt{5}}{2}$ (the golden ratio). - Relationship to the Golden Ratio:
$\displaystyle \lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \varphi$. - Divisibility:
$F_k$ divides $F_{nk}$ for any integer $n$. - Identity Examples:
$F_{m+n} = F_{m-1}F_n + F_mF_{n+1}$ and $F_{2n} = F_n L_n$, where $L_n$ is the n‑th Lucas number.
Applications
- Computer Science: Algorithms for dynamic programming, recursive function analysis, and data structures (e.g., Fibonacci heaps).
- Mathematical Modeling: Population dynamics, branching processes, and phyllotaxis (arrangement of leaves).
- Finance: Technical analysis (Fibonacci retracement levels).
- Art and Architecture: Proportional design based on the golden ratio derived from the sequence.
Extensions and Generalizations
- Generalized Fibonacci Sequences (Lucas Sequences): Defined by the same recurrence with arbitrary initial terms.
- k‑step Fibonacci Numbers (Tribonacci, Tetranacci, etc.): Each term is the sum of the preceding $k$ terms.
- Modulo Sequences: The sequence taken modulo $m$ exhibits periodic behavior (Pisano periods).
References
- Koshy, T. Fibonacci and Lucas Numbers with Applications. Wiley, 2001.
- Sagan, H. Fibonacci Numbers and the Golden Ratio. J. Math. Educ. 81, 2008.
- Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Ellis Horwood, 1989.
This entry provides a concise, factual overview of the Fibonacci sequence as commonly defined in mathematical literature.
