bibkey: oeis2025a084068 authors: OEIS Foundation Inc.; Joseph A. Stocke year: 2025 title: OEIS A084068 gcd conjecture doi: null url: https://oeis.org/A084068 claim: The July 28, 2025 formula conjectures that A084068(n) equals gcd(A001108(n), A001109(n)). strata_touched:

• D5/S1/Recurrence/PellCompanionGcd license: citation-only triage: anchor

OEIS A084068 gcd conjecture

The entry was inspected on September 8, 2026. Its formulas still label Stocke’s gcd assertion as a conjecture. The separate conjecture about the first occurrence of n in A348295 is not addressed here.

Use P(0)=0, P(1)=1 and Q(0)=Q(1)=1, with both sequences satisfying R(n+2)=2R(n+1)+R(n). These are A000129 and A001333 respectively.

The source identification is explicit in the inspected OEIS entries:

• A084068: R. J. Mathar’s October 15, 2021 formula gives P(n) for even n and Q(n) for odd n.
• A001108: the Darin Stephenson and Alan Koch comment gives 2P(n)^2 for even n and Q(n)^2 for odd n. It also gives the coupled recurrence P(n+1)=P(n)+Q(n), Q(n+1)=P(n+1)+P(n).
• A001109: Lekraj Beedassy’s April 23, 2003 comment, edited by Jon E. Schoenfield on May 4, 2014, gives A001333(n)*A000129(n).

These statements verify the normalization used in the Lean theorem. The proof derives coprimality and oddness by induction, then extracts the common factor in each gcd. Documentary identification with the external entries is distinct from the kernel verification of the normalized formula.

Verified locator

• https://oeis.org/A084068 (Stocke gcd conjecture and Mathar parity formula).
• https://oeis.org/A001108 (Stephenson and Koch comment).
• https://oeis.org/A001109 (Beedassy product formula).
• Internal-format pages inspected: https://oeis.org/A084068/internal, https://oeis.org/A001108/internal, https://oeis.org/A001109/internal.