  Dynamical sampling deals with frames of the form
$\{T^n\varphi\}_{n=0}^\infty$, where $T \in B(\mathcal{H})$ belongs to certain
classes of linear operators and $\varphi\in\mathcal{H}$. The purpose of this
paper is to investigate a new representation, namely, Fibonacci representation
of sequences $\{f_n\}_{n=1}^\infty$ in a Hilbert space $\mathcal{H}$; having
the form $f_{n+2}=T(f_n+f_{n+1})$ for all $n\geqslant 1$ and a linear operator
$T :\text{span}\{f_n\}_{n=1}^\infty\to\text{span}\{f_n\}_{n=1}^\infty$. We
apply this kind of representations for complete sequences and frames. Finally,
we present some properties of Fibonacci representation operators.
