  The ruler function or the Gros sequence is a classical infinite integer
sequence that is underlying some interesting mathematical problems. In this
paper, we provide four new problems containing this type of sequence: (i) a
demographic discrete dynamical automata, (ii) the middle interval Cantor set,
(iii) the construction by duplication of polygons and (iv) the horizontal
visibility sequence at the accumulation point of the Feigenbaum cascade. In all
of them, the infinte sequence is obtained by a recursive procedure of
duplication. The properties of the ruler sequence, in particular, those
relating to recursiveness and self-containing, are used to get a deeper
understanding of these four problems. These new representations of the ruler
sequence could inspire new studies in the field of discrete mathematics.
