For any integer

, each palindrome of n induces a circulant graph of order n. It is known that for each integer

, there is a one-to-one correspondence between the set of (resp. aperiodic) palindromes of n and the set of (resp. connected) circulant graphs of order n (cf. [2]). This bijection gives a one-to-one correspondence of the palindromes

with

to the connected circulant graphs. It was also shown that the number of palindromes

of n with

is the same number of aperiodic palindromes of n. Let

(resp.

) be the number of aperiodic palindromes

of n with

(resp.

). Let

(resp.

) be the number of periodic palindromes

of n with

(resp.

). In this paper, we calculate the numbers

,

,

,

in two ways. In Theorem 2.3, we

recurrence relations for

,

,

,

in terms of

for

and

. Afterwards, we nd formulae for

,

,

,

explicitly in Theorem 2.5.