the length of the longest sequence of consecutive fs-double squares in a word

A square is a concatenation of two identical words, and a word w is said to have a square yy if w can be written as xyyz for some words x and z. it is known that the ratio of the number of distinct squares in a word to its length is less than two, and any location of a word could begin with two distinct squares which are appearing in the word for the last time. a square whose first location starts with the last occurrence of two distinct squares is an fs-double square. we explore and identify the conditions under which a sequence of locations in a word starts with fs-double squares. we first find the structure of a word that begins with two consecutive fs-double squares and obtain its properties that enable us to extend the sequence of fs-double squares. it is proved that the length of the longest sequence of consecutive fs-double squares in a word of length n is at most n/7. we show that the squares in the longest sequence of consecutive fs-double squares are conjugates.