The complexity of -words of the form

Abstract: Let denote the number of -words of the form  with gap and denote the number of -words of the form  with length and gap , where is the length of the word . [S. Brlek, A. Ladouceur, A note on differentiable palindromes, Theoret. Comput. Sci. 302 (2003) 167–178] proved that -palindromes are characterized by the left palindromic closure of the prefixes of the well-known Kolakoski sequences and revealed an interesting perspective for understanding some of the conjectures. In fact, they found all infinite -palindromes and established for all , where is the set of positive integers. [Y.B. Huang, About the number of -words of form , Theoret. Comput. Sci. 393 (2008) 280–286] obtained for all and , and gave all -words of the form with gap less than 5, which imply , and . In this paper, we prove the following intriguing results: (1) If and then the first and last letters of the word are the same; (2) ; (3) For every positive integer , there exists a positive integer such that for all , if then if is odd and if is even, which would help us understand better the complexity of finite -words of the form . Moreover, we provide all twenty eight -words of the form . [Copyright &y& Elsevier]