Asymptotic number of isometric generalized Fibonacci cubes

For a binary word ff, let Qd(f)Qd(f) be the subgraph of the dd-dimensional cube QdQd induced on the set of all words that do not contain ff as a factor. Let GnGn be the set of words ff of length nn that are good   in the sense that Qd(f)Qd(f) is isometric in QdQd for all dd. It is proved that limn→∞|Gn|/2nlimn→∞|Gn|/2n exists. Estimates show that the limit is close to 0.080.08, that is, about eight percent of all words are good.