Sur la suite des nombres de la forme qn1 +⋯+ qnk

Let r ≧ 1 be an integer and denote by ξr the root of the polynomial Xr+1 - Xr -⋯- 1 belonging to ]1, 2[. Let (yn (ξr))n be the strictly increasing sequence composed by the real numbers of the form ξrn1 +⋯+ ξrnk, the ni's being distinct non-negative integers. For n ≧ 0, set un(ξr) = yn+1(ξr) - yn(ξr). The purpose of this note is to give a very precise description of the sequence (un(ξr))n, and to show that this is the fixed point of a substitution.