On the Numbers of Palindromes

For any integer n ≥ 2, each palindrome of n induces a circulant graph of order n. It is known that for each integer n ≥ 2, 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 σ with gcd(σ) = 1 to the connected circulant graphs. It was also shown that the number of palindromes σ of n with gcd(σ) = 1 is the same number of aperiodic palindromes of n. Let an (resp. bn) be the number of aperiodic palindromes σ of n with gcd(σ) = 1 (resp. gcd(σ) ̸= 1). Let cn (resp. dn) be the number of periodic palindromes σ of n with gcd(σ) = 1 (resp. gcd(σ) ̸= 1). In this paper, we calculate the numbers an, bn, cn, dn in two ways. In Theorem 2.3, we find recurrence relations for an, bn, cn, dn in terms of ad for d|n and d ̸= n. Afterwards, we find formulae for an, bn, cn, dn explicitly in Theorem 2.5.