Ultimate periodicity of all greedy strict 3-sumfree sequences

Prove that every greedy strict 3-sumfree sequence S_{f,g,h}, with integers f<g<h, is ultimately periodic.

Background

A greedy strict 3-sumfree sequence S_{f,g,h} is generated from three initial values f<g<h by repeatedly choosing the least subsequent integer that is not the sum of three distinct earlier entries. The paper proves ultimate periodicity for broad parameterized families, including sequences of the form S_{f,g,g+d} under explicit inequalities.

The authors report extensive computational evidence, but their general theorem leaves finitely many exceptional parameter values for each fixed pair of parameters and does not establish the conjecture in full. The conjecture is the paper’s principal unresolved periodicity question for t=3.

References

We did very extensive experiments to provide evidence for the next conjecture. Every $3$-sumfree sequence $S_{f,g,h}$ is ultimately periodic.

— On $t$-sumfree sequences  (2609.16843 - Berkel et al., 15 Sep 2026) in Conjecture (folklore), Section 1, following Theorem \ref{thm:k=3d>1short}

The latter cases are those with values $k=p=-1$ and $m=0$ in the table. For these 12 cases we have calculated at least 500000 entries of $S_{f,g,h}$.

— On $t$-sumfree sequences  (2609.16843 - Berkel et al., 15 Sep 2026) in Section 5, paragraph introducing Table \ref{dat}