Wythoff-Fibonacci Sequences and a Perturbed Greedy Almost-involution

We introduce the lower and upper Wythoff-Fibonacci sequences, obtained from the classical Wythoff sequences by a Fibonacci correction. Specifically, if we put $$\epsilon (j)=\{\begin{array}{ll}(-1{)}^k,& \text{if }j=F_k\text{ for some }k\\ 0,& \text{in other case}\end{array},$$ where $F_k$ is the $k$-th Fibonacci number, then we define the general terms of the lower and upper Wythoff-Fibonacci sequences by $$LWF(n)=\{\begin{array}{ll}1,& \text{if }n=1,\\ 3,& \text{if }n=2,\\ a(n)+\epsilon (n),& \text{if }n\ge 3.\end{array}$$ and $$UWF(n)=\{\begin{array}{ll}2,& \text{if }n=1,\\ b(n)+\epsilon (n),& \text{if }n\ge 2,\end{array}$$ respectively. We show that these sequences partition the set of natural numbers and use them to give an explicit formula for a sequence $q_j^{\star }$, defined from a greedy construction studied by the first author and his coauthors in a previous paper, but with the additional condition that $q_1^{\star }=3$, instead of being defined by the greedy rule. This sequence is a permutation of the set of non-negative integers and has the property that every integer appears exactly once in the sequence of differences $q_j^{\star }-j$. We prove that $q_{q_j^{\star }}^{\star }=j\mathrm{\forall }j\ge 5$, so that $q_j^{\star }$ is an almost-involution. We also give another greedy algorithm generating $q_j^{\star }$.