slug: hanna-a382318-a382319-lambert-cube-mod-three
bibkey: hanna2025a382318
doi: null
url: https://oeis.org/A382318
triage: theorem
motivation_gids:
  - D5/S1/Recurrence/Invariants/HannaLambertCubeModThree.result

Hanna’s A382318 congruence modulo three

Problem

Paul D. Hanna’s OEIS A382318, Comments (unpaginated), states:

Conjectures: a(3n) == A382319(n) (mod 3) for n >= 0; a(3n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 1.

The source equations are $A=X+L(A{)}^3$ and $B=X(1-X{)}^{-1}+L(B^3)$, where $L(F)={\sum }_{k\ge 1}F(X^k)$ and both series have zero constant term. At degree $m$, $L(F)$ has coefficient ${\sum }_{1\le k\le m,k\mid m}[X^{m/k}]F$. The empty degree-zero sum is zero. The cube in the second equation is taken before Lambert summation. The source is recorded in the Library note. Preregistration: #18861.

Statement echo

The first congruence is $a(3n)\equiv b(n)(\mathrm{mod}3)$ for every natural $n$. The other two are $a(3n+1)\equiv a(3n+2)\equiv 0(\mathrm{mod}3)$ for $n\ge 1$. No source clause is dropped. The complete proposition is displayed at both claim and result in the Blueprint.

Motivation

The congruence links two implicit generating functions rather than two independently chosen coefficient lists. The coefficientwise Lambert relation and the identity $A=X+B^3$ identify the common realization needed for Frobenius.

Gap

The source and literature screen recorded in #18861 found no published proof or refutation in the entry, its links, the searched OEIS-proof papers and formal conjecture catalogues. Completeness and continued external open status are ASSUMED-UNVERIFIED. The delivered result is a settlement under the literal preregistered rational power-series conventions.

Route

The first module constructs the integral fixed point of $F\mapsto X+L(F{)}^3$ by coefficient stabilization. The contraction of agreement below a degree gives existence and uniqueness. Compatibility with coefficient ring maps identifies every rational solution with the integral one. Linearity of $L$ and $L(X)=X(1-X{)}^{-1}$ give $B=L(A)$ and $A=X+B^3$. Cubic Frobenius modulo three then gives $A\equiv X+B(X^3)$. The proof uses coefficient agreement Agree and agree_cube from CubicFifteenSubstitutionParity, and cube_congr_subst_three from CubicTernarySubstitutionParity. These are actual elaborated dependencies.

Falsifier

A solution pair with a nonintegral relevant coefficient or a violated residue would contradict result. Interpreting the infinite sum with a nonzero constant coefficient would change the source-to-formalization bridge and is excluded by the source offsets and the explicit predicate hypotheses.

Evidence

D5.S1.Recurrence.Invariants.HannaLambertCubeModThree.result proves claim. The two delivered modules carry admission bases open-problem-resolution (#18861; Proved) and open-problem-resolution (#18862; Proved) respectively. Their axiom closures are contained in $\{\mathrm{propext},Classical.choice,Quot.sound\}$.

The four-slot escape audit is unfinished for both public results: issue #19096. The source propositions mix solution existence, universal solution-pair hypotheses and integer witnesses. No compiled registration reconstructing this whole telescope, observational dependence, variation and sensitivity is delivered. No declared_validated registration is claimed.

Experiment

Published checker and data. Command: python3 check.py 300, run in that directory; exit 0, RESULT PASS. The pinned check.py SHA-256 is 319ee58b884641c875833eb3e830906af72d593ab73e1d6b0da6ff155cd352f3. The checked A382318 and A382319 coefficients match their b-files through index 300; $L(A)=B$, $A=X+B^3$ and $B^3\equiv B(X^3)(\mathrm{mod}3)$ hold through degree 300. The paired congruence is checked for $3n\le 300$; the two zero residues are checked for $n\ge 1$ and $3n+2\le 300$. These finite computations do not establish an unbounded statement.

Triage

What the settlement shows

Proved, kernel-checked. HannaLambertCubeModThree.result gives the full conjecture with existence and universal solution-pair quantification. The first module’s private solution_fixed, fixed_unique, solution_link and integral_pair establish the fixed-point mechanism, uniqueness, $B=L(A)$, $A=X+B^3$, and integrality used in this conclusion. These helpers are supporting proof facts, not additional public settlements.

Proved in prose; general-prime Lean formalization open. For a prime $p$, replace the cube by the $p$th power. Every $L(F)$ has zero constant coefficient. If $F$ and $G$ agree below degree $d$, the factorization of $L(F{)}^p-L(G{)}^p$ shows agreement below $d+p-1$: one factor agrees below $d$, and each of the remaining $p-1$ factors has order at least one. Since $p\ge 2$, iteration stabilizes each coefficient, giving a unique fixed point over $\mathbb{Z}$ and then over $\mathbb{Q}$. Linearity of $L$ gives a solution $B=L(A)$ of $B=X(1-X{)}^{-1}+L(B^p)$. Conversely, that equation gives $L(X+B^p)=B$, so $X+B^p$ is a solution of the first equation. Its uniqueness makes this correspondence unique. Over ${\mathbb{F}}_p$, Frobenius gives $B(X{)}^p=B(X^p)$, hence $a(pn)\equiv b(n)(\mathrm{mod}p)$ for all $n\ge 0$, and $a(pn+r)\equiv 0(\mathrm{mod}p)$ when $n\ge 1$ and $0<r<p$. This is a mathematical extension of the mechanism, not a Lean theorem of these modules and not an assertion about the experiment’s tested scope.

Explanation. The exceptional term $X$ accounts for $a(1)=1$; the zero-residue assertion in the source therefore starts at $n=1$. The source’s reciprocal congruence is settled by symmetry in #18862. No other source result depending on the conjecture was identified in the recorded screen. The entry’s asymptotic formula remains open here, since coefficient residues do not supply an asymptotic estimate.

ASSUMED-UNVERIFIED

The literature screen is not exhaustive. Continued external open status, completeness of the search, and the source-to-formalization interpretation of the OEIS infinite sums are not kernel assertions. The general-prime extension above is a prose proof, with its Lean formalization open.