The A068012 subset sum doubling conjecture

Let a(n) count subsets of {1,...,n} whose element sum is zero modulo six. David A. Corneth conjectured on September 13, 2025 that a(n)=2a(n-1) whenever n>2 and three does not divide n-1.

This is a first-tier recent OEIS conjecture. Direct inspection on September 8, 2026 found the exact statement still labelled Conjecture. The entry's finite verification through 2000 is evidence only, not a general proof.

The implementation brief reports no independent proof or preprint in its searched public indexes. The obligation is an identity for all eligible n, not an extension of the finite verification window.

Write C(m,r) for the count at residue r modulo six. Splitting at the last element gives C(n,r)=C(n-1,r)+C(n-1,r-n). Splitting instead at element three shows C(m,r)=C(m,r+3) for every m at least three.

For m congruent to one modulo three, the total sum m(m+1)/2 is one modulo three. Complementation maps residue zero to the total-sum residue, which is either one or four modulo six. The three-periodicity identifies both with residue one. Thus C(m,0)=C(m,1) for m at least four in that phase. Taking m=n-1 now gives the doubling identity in each permitted residue class; the boundary index n=3 is checked inside the general proof.

The brief's group-ring argument for the second lemma yields the same equality. Projection to modulo three sums the two lifts; three-periodicity makes this twice either lift, and cancellation of two recovers the result.

An n>2 with three not dividing n-1 and a(n) different from 2a(n-1) would contradict the theorem about the defined subset count.

• Module: D5/S1/Recurrence/Parity/SubsetSumModSixDoubling.lean.
• Theorem: subset_sum_mod_six_doubling.
• Count: powerset of Finset.Icc 1 n, filtered by its ZMod 6 sum.
• Auxiliary results: count_succ, count_three_periodic, count_zero_eq_one.
• There is no finite cutoff in any of the public theorem statements.

theorem. The exact universal doubling assertion is proved. No converse or resolution of the entry's other formulas is asserted.

First-publication priority is not established. The literature searches do not exclude private or unindexed proofs. The identification with OEIS is documentary; the kernel verifies the explicitly defined mathematical count.