3415. Palindrome

Alice like strings, especially long strings. For each string, she has a special evaluation system to judge how elegant the string is. She defines that a string $S[1..3n-2]$ is one-and-half palindromic if and only if it satisfies $S[i]=S[2n-i]=S[2n+i-2]$ $(1\le i\le n)$. For example, abcbabc is one-and-half palindromic string, and abccbaabc is not. Now, Alice has generated some long strings. She ask for your help to find how many substrings which is one-and-half palindromic.