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 more

Closed-form expression
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. … See more

Combinatorial proofs
Most identities involving Fibonacci numbers can be proved using combinatorial … See more

Infinite 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 more

A 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 more

The 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 more