Consider the Fibonacci words $B_n$:

- $B_1 = 1$
- $B_2 = 10$
- $B_3 = 101$
- $B_4 = 10110$
- $B_5 = 10110101$

(start with $B_1=1$, and go from $B_n$ to $B_{n+1}$ by replacing every occurence of $1$ in $B_n$ with $10$ and every occurence of $0$ with $1$).

The word $B_n$ has $f_n$ digits, the $n$-th Fibonacci number.

For a given integer $\alpha \geq 2$ and $0 \leq j < \alpha$, denote by $$ R_n(\alpha,j) = \#\bigl\{1 \leq i \leq f_n \mid B_n(i)=1 \,\text{ and }\, i\!\!\!\!\pmod\alpha=j \bigr\} $$ the number of integers $i$ in $\{1, \ldots, f_n\}$ such that $i$ is a "rabbit integer" (I mean $B_n(i)=1$) and $i$ is congruent to $j$ modulo $\alpha$.

Does one know an asymptotic estimate of $R_n(\alpha,j)$ when $n \to \infty$ ?