First-occurrence positions for Fibonacci-number differences

Prove that for the Fibonacci word, the first starting positions of maximum-length monochromatic arithmetic progressions satisfy i(F_{2n+1})=F_{2n+3}−2 and i(F_{2n})=F_{4n}−1.

Background

For each difference d, the function i(d) records the earliest position at which a maximum-length monochromatic arithmetic progression of difference d begins. The paper derives several exact formulas for families such as d=F_n−1, F_n+1, and F_n+2.

For the important remaining family d=F_n, the authors observe empirical evidence for a simple parity-dependent formula. Because A(F_n) is not eventually constant, their automatic-theorem-proving approach cannot directly verify the claim, leaving the conjecture unresolved.

References

Empirical evidence generated by suggests making the following Conjecture \ref{CONJ:idfn}. Unfortunately, because A(F_n) is not eventually constant, as will be shown in the next section, we once again run into the limitation of 's inability to handle the multiplication of two variables.

— Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words  (2501.05830 - Joshi et al., 10 Jan 2025) in Conjecture 3, subsection “Investigating MAPs in the Fibonacci word in Walnut”