Gray codes for Fibonacci q-decreasing words

A length n binary word is q-decreasing, q?1, if every of its maximal factors the form 0a1b satisfies a=0 or q?a>b. In particular, in 1-decreasing words run 0s immediately followed by a strictly shorter 1s. We show constructively that these are bijection with having no occurrences 1q+1, and thus they enumerated (q+1)-generalized Fibonacci numbers. give some enumerative results reveal similarities between q-decreasing 1q+1 terms frequency 1-bits. second part our paper, we provide an efficient exhaustive generating algorithm for lexicographic order, any existence 3-Gray codes explain how Gray can be obtained. Moreover, construction more restrictive 1-Gray code words, which particular settles conjecture stated recently context interconnection networks E?ecio?lu Irši?.