Asymptotic sum for the discrepancy in Cloitre’s Hofstadter-sequence conjecture

Establish that there exists a constant C, approximately equal to 0.53, such that the sum from i=1 to t of h(i) - floor(alpha^{-1} i) is asymptotic to Ct, where h is the Narayana-representation sequence identified with Hofstadter’s H-sequence and alpha is the real root greater than 1 of X^3 = X^2 + 1.

Background

The paper defines h(i) from the Narayana representation of i and proves that h(i) equals Hofstadter’s H-sequence. It also proves Cloitre’s inequality H(i) - floor(alpha{-1} i) in {0,1}.

A further conjecture concerns the cumulative frequency of the values of this discrepancy. The paper explicitly states that its methods are insufficient to prove the claimed asymptotic relation, so the problem remains unresolved within the paper.

References

Cloitre conjectures in https://oeis.org/#1{A082401} that $\sum_{1 \leq i \leq t} (h(i)- \lfloor \alpha{-1} i \rfloor) \sim Ct$ for some constant $C$, approximately $0.53$. The methods in this paper do not seem strong enough to prove this.

— The Narayana Morphism and Related Words  (2503.01026 - Shallit, 2 Mar 2025) in Section “Other sequences based on Narayana representation”, remark following the proof of the conjectured inequality for H(i)