Periodic characterization of the greedy 3-subsumfree sequences S_{1,g,g+1}

Establish, for every integer g≥2, that the greedy 3-subsumfree sequence S_{1,g,g+1} consists exactly of the exceptional values 1, 2g+1, and 6g+1 together with the positive integers whose residues modulo 10g+3 lie in {g,g+1,…,2g}∪{6g+2,6g+3,…,7g+1}, and consequently is eventually periodic with period 2g+1 after its first g+4 entries.

Background

The paper studies greedy sequences that exclude sums of three distinct earlier entries. It proves explicit modular characterizations for S_{1,2,3}, S_{1,3,4}, and S_{1,4,5}, motivating a parametrized extension to all g≥2.

Walnut cannot directly verify the universally quantified parametrized statement because its multiplication operator cannot be applied to two variables. The authors report successful checks for g from 2 through 10, but leave the general conjecture unresolved.

References

In fact we believe the following generalized conjecture to hold. \begin{conjecture}\label{conj_g} For every $g\geq2$ the greedy 3-subsumfree sequence $S_{1,g,g+1}$ is characterized as follows:

— Using Walnut to solve problems from the OEIS  (2503.04122 - Bosma et al., 6 Mar 2025) in Section 4, Conjecture \ref{conj_g}