bibkey: bala2022a002000 authors: N. J. A. Sloane; Peter Bala year: 2022 title: “OEIS A002000, the cubic recurrence a(n+1) = a(n)(a(n)^2 - 3) from 7, with Bala’s 3-adic congruence conjecture” doi: null url: https://oeis.org/A002000 claim: “%N a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 7. %F Conjecture: a(n+1) == a(n) (mod 3^(n+r+2)) for n >= r.” strata_touched:

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

OEIS A002000

The associated theorem proves Bala’s conjectured congruence for every natural n and every natural r <= n. The sequence is integer-valued and defined by the NAME recurrence with initial value 7. Congruence is expressed as divisibility of the integer difference by 3^(n+r+2). Only the Conjecture line is settled; the Lucas representation, the 3-adic limit, and the product formula are not claimed.

Verified locator

• URL: https://oeis.org/A002000
• Source transcription and literature readings: 2026-09-15; ASSUMED-UNVERIFIED.
• %N (verbatim): a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 7.
• %O (verbatim): 0,1
• %A (verbatim): _N. J. A. Sloane_
• Settled %F line (verbatim), inside Peter Bala’s Nov 15 2022 (Start) … (End) block: Conjecture: a(n+1) == a(n) (mod 3^(n+r+2)) for n >= r.
• Context in the same block (verbatim): a(n) = Lucas(4*(3^n)).
• Context in the same block (verbatim): a(n+1) == a(n) (mod 3^(n+1)) (a particular case of the Gauss congruences for the Lucas numbers).
• Context in the same block (verbatim): The least positive residue of a(n) mod(3^n) = 3^n - 2 = A058481(n). In the ring of 3-adic integers the limit_{n -> oo} a(n) exists and is equal to -2.

According to these readings, Bala’s own block proves only the weaker mod 3^(n+1) congruence through the Gauss congruences for Lucas numbers, and no proof of the 3^(n+r+2) strengthening was found in the searched surfaces.