The hardest part, was getting my brain to refocus on this "Special Palindrone" definition. Originally I was coding for a normal palindrone, which is VASTLY different and more complicated. Once I wasted time on that and re-read the problem, the solution wasn't too difficult.

My Solution: C#

    static long substrCount(string s)
    {
        long retVal = s.Length;

        for (int i = 0; i < s.Length; i++)
        {
            var startChar = s[i];
            int diffCharIdx = -1;
            for (int j = i + 1; j < s.Length; j++)
            {
                var currChar = s[j];
                if (startChar == currChar)
                {
                    if ((diffCharIdx == -1) ||
                       (j - diffCharIdx) == (diffCharIdx - i))
                        retVal++;
                }
                else
                {
                    if (diffCharIdx == -1)
                        diffCharIdx = j;
                    else
                        break;
                }
            }
        }
        return retVal;
    }