On the structure of the complement $\overline{\text{Mfib}}$ of the set $\text{Mfib}$ of fibbinary numbers in the set of positive natural numbers

The set $\text{Mfib}$ of fibbinary numbers is defined via a bijection between the set $\text{BB}N$ of natural numbers and $\text{Mfib}$. Since the elements of $\text{Mfib}$ do not exhaust $\text{BB}N$, the structure of the complement $\overline{\text{Mfib}}$ of $\text{Mfib}$ in $\text{BB}N$ is of interest. An explicit expression $\overline{\text{Mfib}}={\bigcup }_{k\ge 1}^{\mathrm{\infty }}{\mathrm{\Phi }}_k$ is obtained in terms of certain well-defined sets ${\mathrm{\Phi }}_k,k\ge 1$. The key to its proof lies in first considering the odd numbers involved in this statement: a general treatment, with full justification, of the binary representations of the odd numbers is developed, and exploited in showing the expression quoted for $\overline{\text{Mfib}}$ to be correct. The main results of the article can also be viewed as providing partitions of the set $\text{BB}N$ of natural numbers, and also of its subset of odd numbers, that follow from the introduction of the set $\text{Mfib}$, and of its subset of odd integers.