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The ruler function or the Gros sequence is a classical infinite integersequence that is underlying some interesting mathematical problems. In thispaper, we provide four new problems containing this type of sequence: (i) ademographic discrete dynamical automata, (ii) the middle interval Cantor set,(iii) the construction by duplication of polygons and (iv) the horizontalvisibility sequence at the accumulation point of the Feigenbaum cascade. In allof them, the infinte sequence is obtained by a recursive procedure ofduplication. The properties of the ruler sequence, in particular, thoserelating to recursiveness and self-containing, are used to get a deeperunderstanding of these four problems. These new representations of the rulersequence could inspire new studies in the field of discrete mathematics.

On the ubiquity of the ruler sequence