slug: oeis-a381365-scaled-bilateral-product-quadratic-mod-nine bibkey: hanna2025a381365 doi: null url: https://oeis.org/A381365 triage: theorem motivation_gids:

• D5/S1/Recurrence/Bilateral/ScaledBilateralProductModNine

The A381365 scaled bilateral product congruence

Problem

This dossier cites its own Library note, Library/ArithSums/hanna2025a381365.md, for the A381365 conjecture. The two entry statements describe instances of one family:

OEIS A381364, Paul D. Hanna, Feb 21 2025, NAME:

G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^nA(x)^n * (A(x)^n + 2x)^(n-1) * (x^n + 2*A(x))^(n-1).

COMMENT:

Conjecture: for n > 0, a(n) == 6 (mod 9).

OEIS A381365, Paul D. Hanna, Feb 21 2025, NAME:

G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^nA(x)^n * (A(x)^n + 2x)^(2n-1) * (x^n + 2A(x))^(2*n-1).

COMMENT:

Conjecture: for n > 0, a(n) == 6 (mod 9).

Motivation

This is a first-tier OEIS conjecture from 2025. The mathematical target is a universal congruence for the normalized generating series. KPI = open problems resolved; no repository cumulative count is asserted.

Gap

The supplied search report found no proof: the search seat read the OEIS entry and revision history on 2026-09-09 and performed identifier searches on arXiv, MathOverflow, and GitHub. This Stage-B seat had no network access and did not repeat those searches. The reported gap is a proof of the universal congruence from the literal bilateral equation, including the negative indices.

Route

The escape witness is the general theorem hanna_conjecture_general, covering every c ≥ 1, rather than the two named instances: the unique normalized integer solution of 1/3 = Σ_{n∈ℤ} xⁿ·Aⁿ·(Aⁿ + 2x)^{cn−1}·(xⁿ + 2A)^{cn−1} has a(c,n) ≡ 6 (mod 9) for every n > 0. In the checked source’s parameter convention, A381364 is c = 1 and A381365 is c = 2.

The literal Laurent summand is retained in laurentTerm over LaurentSeries ℚ with integer exponents, including negative indices. Thus the formalized equation is the OEIS equation, rather than a paraphrase. The finite-window theorem and pairing of opposite indices give a well-defined integral coefficient construction: the summand has order at least n for n ≥ 1 and order at least c·k² for n = −k, k > 0. Each coefficient therefore depends on finitely many indices; no infinite sum object or summability assumption is used. Formal power series and Laurent series remain the ambient objects. Pairing proves the nonzero remainder even, giving H = 2J. An integral contracting iteration has a fixed point with a c n := coeff n (approximation c (n+1)).

The normalized form (1 + 2x)(1 + 2A)(1 − 3H) = 3 yields A − (1 − 3xG) = 9(1 + 2B)J, where G = (1 + 2x)⁻¹. Since coeff m G = (−2)^m ≡ 1 (mod 3), every positive coefficient is 6 (mod 9). Existence uses generating_equation; generating_unique identifies every normalized integer solution with the constructed series.

The actual IMPORTED and frozen prerequisite is D5/S1/Recurrence/Bilateral/BilateralProductModFour, statement_id sha256:e80c04dabc53956b6912a1f482e052ba0c5fb1bb30d501daf127d6ae3b41e406; the module header therefore has generality I. Its public statements concern the unscaled family; the implementation adapted the private agreement and embedding arguments rather than reusing them verbatim. The brief’s requested prerequisite D5/S1/Recurrence/Bilateral/ScaledBilateralProductModNine is the target itself, not its import. Its requested Golden/Frozen/state/D5/S1/Recurrence/Bilateral/ScaledBilateralProductModNine.lean.json was absent in the Stage-B inspection, so no statement_id for that target can be quoted. This dependency discrepancy is recorded as a plan deviation; no self-import or unverified freeze is asserted.

Falsifier

A positive index n in the normalized integer solution for A381365 (c = 2) whose coefficient is not congruent to 6 modulo 9 would falsify this entry’s assertion. More generally, such an index for any c ≥ 1 would falsify the escape theorem. The orchestrator’s exact finite check is supporting evidence only and cannot exclude every counterexample index.

Evidence

• Lean module: D5/S1/Recurrence/Bilateral/ScaledBilateralProductModNine.lean.
• Entry theorem: hanna_conjecture_a381365; the general escape theorem is hanna_conjecture_general. The source has no declaration named hanna_conjecture.
• Companions: finite_window, polynomial_form (the normalized form (1 + 2x)(1 + 2A)(1 − 3H) = 3), generating_equation, and generating_unique.
• Axiom boundary: std3 (propext, Classical.choice, Quot.sound); single-file verification output is recorded in build/op-a381364-stage-b/lean.log.

The orchestrator supplied readings attributed to results/verify-r21.out: for both c = 1 and c = 2, coefficients were integral through n < 24, the computed coefficients reproduced the published DATA exactly (1, 6, 69, 1185, 25971, 638664, 16870146 and 1, 6, 267, 13686, 850848, 58650900, 4328042982, respectively), the residual of the defining relation was identically zero through degree 23, and a(n) ≡ 6 (mod 9) held at every index 1..23, with zero violations in both cases. The orchestrator solved the defining relation in exact rational arithmetic, rather than reading coefficients off a printed recurrence. These are supporting checks, not proof; the referenced output file was absent from this worktree during Stage-B inspection, so this seat did not independently verify those readings.

Triage

theorem. The general formal statement specializes to this entry’s conjecture at c = 2. Its source identification is subject to the provenance boundary below.

ASSUMED-UNVERIFIED

The OEIS quotations and entry date were supplied by the orchestrator and copied from the Library notes. The OEIS entry and revision history were read by the search seat, not this seat. The reported literature scope is the OEIS entry and revision history plus identifier searches on arXiv, MathOverflow, and GitHub; it does not establish an exhaustive literature search or first-publication priority. This seat had no network. The numerical readings above are attributed to the orchestrator; their referenced file was not available here. Source-to-Lean identification and the implementation’s adaptation history are not kernel-checked facts. The supplied implementation-seat envelope described an earlier simplified proof and was not used to certify the inspected source or its escape witness.