Structure and growth of R-bonacci words

16 pages, 7 figures, 3 tables ; A Sturmian word of slope q is the cutting sequence of a half-line y = qx. We establish a bijection between sequences of certain prefixes of the Sturmian word of slope q, and the q-decreasing words, which are binary words whose maximal factors of the form 0 a 1 b satisfy q • a > b whenever a > 0. We also show that the number of q-decreasing words of length n grows as Φ(q) n(1+o(1)) , where Φ(1) is the golden ratio, Φ(2) is equal to the tribonacci constant, and that the function Φ(q) is strictly increasing, discontinuous at every positive rational point, and exhibits a nice fractal structure related to the Stern-Brocot tree and Minkowski's question mark function.