First differences of subword complexity for the generalized Allouche–Johnson words

Determine whether, for every integer k >= 1 and all sufficiently large n, the first difference rho_xk(n+1) - rho_xk(n) of the subword-complexity function of the infinite word x_k takes only the values 4k-2 and 4k.

Background

For the family x_k, the paper compares known or established values of the first difference of subword complexity in the cases k=1, k=2, and k=3. Specifically, the first difference takes values in {2,4} for Thue–Morse, in {6,8} for the Fibonacci–Thue–Morse word for sufficiently large n, and in {10,12} for the Allouche–Johnson word.

These cases suggest the general pattern {4k-2, 4k}. The conjecture asks whether this two-value behavior persists for every member of the family once n is sufficiently large.

References

This suggests the following conjecture: The first difference of the subword complexity function, $\rho_{{\bf x}k}(n+1)-\rho{{\bf x}_k}(n)$ for ${\bf x}_k$, for $n$ large enough, takes the values $4k-2$ and $4k$ only.

— The Narayana Morphism and Related Words  (2503.01026 - Shallit, 2 Mar 2025) in Section 11, “Final words”