Fibonacci numbers
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| Fibonacci numbers | |
|---|---|
| Name | Fibonacci numbers |
| Caption | Spiral approximating ratios of consecutive terms |
| Field | Number theory |
| Discovered | 1202 |
| Discoverer | Leonardo of Pisa (Fibonacci) |
| Sequence | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... |
Fibonacci numbers
The Fibonacci numbers form an integer sequence beginning with 0 and 1 in which each subsequent term is the sum of the two preceding terms. Originating in medieval Leonardo of Pisa's Liber Abaci, the sequence has been studied in Pisa and across Europe and appears in problems posed by Mathematics competitions and texts by Édouard Lucas and Johann Heinrich Lambert. The sequence connects to diverse topics such as Lucas numbers, the golden ratio, and algorithms used by Donald Knuth and practitioners at institutions like Bell Labs.
Definition and sequence
The sequence is defined by a recurrence relation with initial conditions F0 = 0, F1 = 1; for n ≥ 2, Fn = F_{n−1} + F_{n−2}. Early expositions appear in manuscripts associated with Leonardo of Pisa and were disseminated through scholarly centers in Florence and Pisa. Variants and indexing conventions are used by authors such as Édouard Lucas and in curricula at University of Cambridge and Massachusetts Institute of Technology. The list begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... and is catalogued in compendia used by Encyclopaedia Britannica and by editors at Oxford University Press.
Mathematical properties
The sequence satisfies linear recurrence relations studied in Carl Friedrich Gauss's era and in modern treatments by researchers affiliated with Princeton University and ETH Zurich. Consecutive terms are coprime and many divisibility properties hold: Fn divides Fm whenever n divides m, a fact discussed in lectures at Harvard University and in papers by Srinivasa Ramanujan observers. The sequence grows exponentially with index n; asymptotic estimates are derived in contexts addressed by Pierre-Simon Laplace and refined in analytic work at Institut des Hautes Études Scientifiques. Parity patterns, periodicity modulo m (Pisano periods), and primality of specific terms have been studied by mathematicians associated with Cambridge University and the American Mathematical Society.
Closed form and identities
A closed-form expression, often attributed in exposition to methods popularized after Binet's work, expresses Fn using powers of the golden ratio and its conjugate; this formula appears in textbooks published by Springer and Cambridge University Press. Numerous identities connect terms: Cassini's identity, d'Ocagne's identity, and summation formulas found in compilations by G. H. Hardy and S. Ramanujan; these identities are used in proofs presented at seminars at École Normale Supérieure and in articles in journals of the American Mathematical Society. Matrix representations using 2×2 matrices yield efficient exponentiation techniques employed in computational work at IBM and in algorithmic texts by Donald Knuth.
Combinatorial and geometric interpretations
Combinatorial interpretations include counting tilings of boards and compositions, topics treated in exercises at University of Oxford and in monographs by Richard Stanley. Geometric constructions relate terms to polygonal structures and continued fractions discussed in treatises by Luca Pacioli and later visualizations held by curators at the Vatican Museums. The Fibonacci spiral approximates logarithmic spirals seen in works exhibited at institutions like the Louvre and analyzed by scholars at the Smithsonian Institution. Interpretations connect to problems in recreational mathematics popularized by publishers such as Dover Publications.
Generalizations and related sequences
Generalizations include the Lucas sequence, k-step Fibonacci sequences (Tribonacci, Tetranacci), and linear recurrences parameterized in research groups at MIT and Caltech. The theory of linear recurrences with constant coefficients links to algebraic frameworks developed by Évariste Galois and tools used in algebraic studies at University of Paris. Related integer sequences are catalogued by contributors to projects at The On-Line Encyclopedia of Integer Sequences and in databases maintained by institutions like Wolfram Research. Multivariate and matrix generalizations appear in work by faculty at Stanford University and in conference proceedings of the International Congress of Mathematicians.
Applications in science and art
Fibonacci numbers appear in algorithmic analyses in computer science laboratories at Bell Labs and Microsoft Research and in models of population growth discussed in ecological studies at Smithsonian Tropical Research Institute. They are used in pseudo-random generators in engineering programs at Georgia Institute of Technology and in signal processing research at ETH Zurich. In art and architecture, proportions related to the golden ratio have been explored in restorations conducted by teams at The British Museum and in design principles taught at École des Beaux-Arts. References to Fibonacci-like patterns appear in botanical studies at Kew Gardens and in music theory work by scholars affiliated with Juilliard School.