The maternal-generation conjecture for the meta-Fibonacci sequence μ(n)

Let $\mu(n)$ be the sequence defined by

$$\mu(n)=\mu(n-1-\mu(n-2))+\mu(n-2-\mu(n-1)),\qquad \mu(1)=\mu(2)=\mu(3)=1.$$

The maternal generation sequence of $\mu(n)$ records the maternal generation associated with each term of the sequence.

Maternal-generation conjecture. The sequence $\mu(n)$ is defined for all positive integers $n$, and its maternal generation sequence is slow-growing. For each $g\geq 3$, the $g$th maternal generation begins at index $2^{g-1}+g$, which is the first occurrence of $2^{g-2}+1$ in $\mu(n)$, and ends at index $2^g+g$, which is the last occurrence of $2^{g-1}$ in $\mu(n)$.

The conjecture captures regularities observed computationally in the first $10^6$ terms, including the generation structure and the narrowing intervals associated with powers of $2$. It asserts both global well-definedness of the recurrence and an exact description of every maternal generation; no proof or resolution is given here.