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slug: oeis-a068012-correction-recurrence bibkey: oeis2025a068012 doi: null url: https://oeis.org/A068012 triage: theorem motivation_gids:

  • D5/S1/Recurrence/SubsetSums/ModSixResidueCounts

Corneth’s second recurrence for OEIS A068012

Problem

Let a(n) count subsets of {1,…,n} whose element sum is zero modulo six. David A. Corneth’s September 13, 2025 FORMULA contribution states:

a(3k+1) = 2a(3k) - 2^(k-1), k >= 1, 3k + 1 = n.

Motivation

This is the second recurrence of a first-tier recent OEIS conjecture selected in the orchestrator’s v3 bind-first brief. Its reported finite verification through 2000 does not prove the universal statement.

Gap

PR #6434 resolved only the doubling recurrence, with the frozen module D5/S1/Recurrence/Parity/SubsetSumModSixDoubling. The correction recurrence requires the exact difference of counts at successive indices. This module imports that owner and uses its C, a, and count_succ directly.

Route

Prove the full residue table for every m >= 3 by natural-number induction from m=3, propagating all six residues with the frozen count_succ. Write q=m div 3. In phase m mod 3=1 the residues 2 and 5 satisfy 6C(m,r)+22^q=2^m, and the other residues satisfy 6C(m,r)=2^m+2^q. In the other phases residues 0 and 3 satisfy 6C(m,r)=2^m+22^q, and the other residues satisfy 6C(m,r)+2^q=2^m.

Apply the table at residue zero for 3k and 3k+1. With k>=1, 2^k=22^(k-1); cancellation gives a(3k+1)+2^(k-1)=2a(3k). This also implies the source subtraction form.

Falsifier

A k>=1 for which a(3k+1)+2^(k-1) differs from 2*a(3k) would contradict the theorem about the frozen subset count.

Evidence

  • Module: D5/S1/Recurrence/SubsetSums/ModSixResidueCounts.lean.
  • Theorem: corneth_step.
  • Full residue table: count_closed_form.
  • Definitions and insertion recurrence belong to the frozen doubling module.
  • Both theorem statements are unbounded; there is no public finite regression.

Triage

theorem. The second recurrence is proved for every k>=1. The doubling result belongs to #6434 and is not claimed as new here.

ASSUMED-UNVERIFIED

The OEIS quotation and attribution are supplied by the orchestrator’s brief and prior source inspection. This Stage-A worker has no network and did not repeat that inspection or third-party literature searches. First-publication priority, exhaustive literature coverage, and OEIS revision history are not established. The source identification is documentary; the kernel checks the explicit mathematical statement. Stage-B emission, Scribe compilation, and repository admission remain for the orchestrator.