  Let ${\cal P}$ be the set of palindromes occurring in the Fibonacci sequence.
In this note, we establish three structures of $\mathcal{P}$ and and discuss
their properties: cylinder structure, chain structure and recursive structure.
Using these structures, we determine that the number of distinct palindrome
occurrences in $\mathbb{F}[1,n]$ is exactly $n$, where $\mathbb{F}[1,n]$ is the
prefix of the Fibonacci sequence of length $n$. Then we give an algorithm for
counting the number of repeated palindrome occurrences in $\mathbb{F}[1,n]$,
and get explicit expressions for some special $n$, which include the known
results. We also give simpler proofs of some classical properties, such as in
X.Droubay, W.F.Chuan and J.Shallit et al.
