Asymptotic number of isometric generalized Fibonacci cubes

For a binary word f , let Q d ( f ) be the subgraph of the d -dimensional cube Q d induced on the set of all words that do not contain f as a factor. Let G n be the set of words f of length n that are good in the sense that Q d ( f ) is isometric in Q d for all d . It is proved that lim n → ∞ | G n | / 2 n exists. Estimates show that the limit is close to 0.08 , that is, about eight percent of all words are good.