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The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence typically begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on12. This sequence was introduced to Western mathematics by Leonardo of Pisa, also known as Fibonacci, in his 1202 book "Liber Abaci"1.
Mathematical Definition
The Fibonacci sequence can be defined by the recurrence relation: [ F(n) = F(n-1) + F(n-2) ] with initial conditions: [ F(0) = 0, \quad F(1) = 1 ]
Code Example
Here is a simple Python function to generate the Fibonacci sequence up to the nth term:
def fibonacci(n):fib_sequence = [0, 1]while len(fib_sequence) < n:fib_sequence.append(fib_sequence[-1] + fib_sequence[-2])return fib_sequence# Example usageprint(fibonacci(10)) # Output: [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]Applications and Properties
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See all on WikipediaFibonacci sequence - Wikipedia
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . Many writers begin the sequence with 0 and 1, although some authors start it … See more
The Fibonacci numbers may be defined by the recurrence relation $${\displaystyle F_{0}=0,\quad F_{1}=1,}$$ and $${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$ for n > 1.
Under some older … See moreClosed-form expression
Like every sequence defined by a homogeneous linear recurrence with constant coefficients, the Fibonacci numbers have a See moreCombinatorial proofs
Most identities involving Fibonacci numbers can be proved using combinatorial … See moreInfinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions. For example, the sum of every odd … See more
India
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. In the Sanskrit poetic tradition, there was interest in … See moreA 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
$${\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}$$ alternatively denoted
which yields See moreThe generating function of the Fibonacci sequence is the power series
$${\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}$$ See moreWikipedia text under CC-BY-SA license Fibonacci Sequence - Definition, List, Formulas and Examples
What is Fibonacci Sequence? The Fibonacci sequence, also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the …
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