  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 at most two
rightmost distinct squares. 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 to generate a sequence of locations in a word that
starts with FS-double squares. We first find the structure of the smallest word
that begins with two consecutive FS-double squares and obtain its properties
that enable 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 $\frac{n}{7}$. We show that the squares in the longest
sequence of consecutive FS-double squares are conjugates.
