  Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating
set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a
vertex in $V - S$. The restrained domination number of $G$, denoted by
$\gamma_r(G)$, is the smallest cardinality of a restrained dominating set of
$G$. Finding the restrained domination number is NP-hard for bipartite and
chordal graphs. Let $G_n^i$ be the family of restrained dominating sets of a
graph $G$ of order $n$ with cardinality $i$, and let $d_r(G_n, i)=|G_n^i|$. The
restrained domination polynomial (RDP) of $G_n$, $D_r(G_n, x)$ is defined as
$D_r(G_n, x) = \sum_{i=\gamma_r(G_n)}^{n} d_r(G_n,i)x^i$. In this paper, we
focus on the RDP of cycles and have, thus, introduced several novel ways to
compute $d_r(C_n, i)$, where $C_n$ is a cycle of order $n$. In the first
approach, we use a recursive formula for $d_r(C_n,i)$; while in the other
approach, we construct a generating function to compute $d_r(C_n,i)$.
