  Let $U_n$ be the set of un-oriented and rational links with crossing number
$n$, a precise formula for $|U_n|$ was obtained by Ernst and Sumners in 1987.
In this paper, we study the enumeration problem of oriented rational links. Let
$\Lambda_n$ be the set of oriented rational links with crossing number $n$ and
let $\Lambda_n(d)$ be the set of oriented rational links with crossing number
$n$ ($n\ge 2$) and deficiency $d$. In this paper, we derive precise formulas
for $|\Lambda_n|$ and $|\Lambda_n(d)|$ for any given $n$ and $d$ and show that
$$ \Lambda_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor
\frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor
\frac{d+1}{2}\rfloor}, $$ where $F_n^{(d)}$ is the convolved Fibonacci
sequence.

Geometric Topology;Combinatorics