On the Numbers of Palindromes

Abstract

For any integer $n\geq 2$, each palindrome of $n$ induces a circulant graph of order $n$. It is known that for each integer $n\geq 2$, 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. \cite{bbkl}). This bijection gives a one-to-one correspondence of the palindromes $\sigma$ with $\gcd(\sigma)=1$ to the connected circulant graphs. It was also shown that the number of palindromes $\sigma$ of $n$ with $\gcd(\sigma)=1$is the same number of aperiodic palindromes of $n$. Let $a_n$ (resp. $b_n$) be the number of aperiodic palindromes $\sigma$ of $n$ with $\gcd(\sigma)=1$(resp. $\gcd(\sigma)\neq 1$). Let $c_n$ (resp. $d_n$) be the number of periodic palindromes$\sigma$ of $n$ with $\gcd(\sigma)=1$ (resp. $\gcd(\sigma)\neq 1$). In this paper, we calculate the numbers $a_n,b_n,c_n, d_n$ in two ways. In Theorem 2.3,we find recurrence relations for $a_n,b_n,c_n, d_n$ in terms of $a_d$ for $d|n$ and $d\neq n$. Afterwards, we find formulae for $a_n,~b_n,~c_n,~d_n$ explicitly in Theorem 2.5.