Infinite families of 2-isometric and not 3-isometric binary words☆

Abstract Let F k be the family of the binary words containing the letter 0 exactly k times. Ilic, Klavžar and Rho constructed an infinite subfamily of 2-isometric and not 3-isometric words in F 2 . Wei and Zhang further found all such words in F 2 . In this paper we find that there exists no 2-isometric and not 3-isometric word in F 3 . For k ≠ 1 , 3 , 4 and 7, we also construct an infinite subfamily of 2-isometric and not 3-isometric words in F k . Based on those results and computer experiments, we conjecture that F 1 , F 3 , F 4 and F 7 are the only families in which there exists no 2-isometric and not 3-isometric word.