The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. In
the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long (L) syllables
that are 2 units of duration, and sh
2-dimensional system of linear difference equations that describes the Fibonacci sequence is
cfw_\displaystyle cfw_F_cfw_k+2 \choose F_cfw_k+1=cfw_\begincfw_pmatrix1&1\1&0\endcfw_pmatrixcfw_F_cfw_k+1 \choose F_cfw_k cfw_\displaystyle cfw_F_cfw_k+2 \choos
The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see binomial
These numbers also give the solution to certain enumerative problems. The most common
such problem is that of counting th
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form solution. It has become known as "Binet's formula", even though it was already known by Abraham de Moivre:
Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of sequences of 1s and 2s that sum to n - 1. This can be taken as the definition of Fn, with the convention tha
The generating function of the Fibonacci sequence is the power series
cfw_\displaystyle s(x)=\sum _cfw_k=0^cfw_\infty F_cfw_kx^cfw_k. s(x)=\sum _cfw_k=0^cfw_\infty F_cfw_kx^cfw_k.
This series is convergent for cfw_\displaystyle |x|<cfw_\frac cfw_1cfw_\var
The Fibonacci numbers are important in the computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
Brasch et al
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called
the Fibonacci sequence, and characterized by the fact that every number after the first two is the
sum of the two preceding ones:
Numerous other identities can be derived using various methods. Some of the most noteworthy are:
Cassini and Catalan's identities
Main article: Cassini and Catalan identities
Cassini's identity states that
The question may arise whether a positive integer x is a Fibonacci number. This is true if and only if one or both of cfw_\displaystyle 5x^cfw_2+4 5x^cfw_2+4 or cfw_\displaystyle 5x^cfw_2-4 5x^cfw_2-4 is a perfect square. This is because Binet's formu
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