A connection between palindromic and factor complexity using return   words
  In this paper we prove that for any infinite word W whose set of factors is closed under reversal, the following conditions are equivalent:   (I) all complete returns to palindromes are palindromes;   (II) P(n) + P(n+1) = C(n+1) - C(n) + 2 for all n, where P (resp. C) denotes the palindromic complexity (resp. factor complexity) function of W, which counts the number of distinct palindromic factors (resp. factors) of each length in W. 
Michelangelo Bucci, Alessandro De Luca, Amy Glen, Luca Q. Zamboni