Ultimate periodicity of all greedy strict 2-sumfree sequences

Prove that for every pair of integers g>f≥1, the greedy strict 2-sumfree sequence S_{f,g} is ultimately periodic.

Background

A greedy strict 2-sumfree sequence S_{f,g} begins with f<g and subsequently selects the least possible integers that are not sums of two distinct earlier entries. The paper proves ultimate periodicity for the substantial family S_{f,f+d} when f≥d+1, but does not cover all parameter pairs.

Computations suggest that every sequence in the two-parameter family is ultimately periodic, motivating the explicit conjecture. The conjecture concerns the characteristic sequence, or equivalently the first-difference sequence, because the original sequence is strictly increasing.

References

Fairly extensive computations do lead us to believe that the following is indeed true. For every $g> f 1$ the 2-sumfree sequence $S_{f, g}$ is ultimately periodic.

— On $t$-sumfree sequences  (2609.16843 - Berkel et al., 15 Sep 2026) in Conjecture in Section 1, following Theorem 2-sumfree family results

The question we would like to address is the main conjecture in this area. Every 2-sumfree sequence is eventually periodic. We have not been able to establish the truth of this conjecture, but instead offer 3 pieces of evidence in support.

— Periodicity conjectures for all 2-sumfree sequences  (2609.18522 - Berkel et al., 16 Sep 2026) in Conjecture 1, Section 1 (Introduction)

We are now ready to state the conjecture for the period length of $S_{f,g}$ for every possible pair of starting values $f, g$ with $g>f>0$; we switch to the use of $f, d=g-f$ as before and write $\PP(d,f)$ for the length of the period of $S_{f, f+d}$. For all $f, d=g-f\geq 1$ the strict, greedy 2-sum sequence $S_{f,g}$ is ultimately periodic with length of the period $\PP(d,f)=\LL_f(d)$.

— Periodicity conjectures for all 2-sumfree sequences  (2609.18522 - Berkel et al., 16 Sep 2026) in Conjecture 2, Section 2 (The conjectures for period lengths)

The main conjecture about the preperiod length is now easily stated, when we use $\QQ(d,f)$ for the length of the preperiod, for each pair of positive integers $f, d=g-f$: For all $f, d=g-f\geq 1$ the strict, greedy 2-sum sequence $S_{f,g}$ is ultimately periodic with preperiod length $\QQ(d,f)=\KK_{r}(d,f)$, where $r\equiv d\bmod 2f$.

— Periodicity conjectures for all 2-sumfree sequences  (2609.18522 - Berkel et al., 16 Sep 2026) in Conjecture 3, Section 3 (The conjectures for preperiod lengths)