Logarithmic-Weight Coefficients Modulo Three

Abstract

OEIS A397349 alternates 1, 2 modulo three at positive indices; its parity conjecture remains open.

The NAME and both conjecture COMMENTS are quoted in hanna2026a397349. Only the modulo-three statement is proved below. The separate statement that a(n) is odd exactly at powers of two is not proved: parity_conjecture is a proposition definition carrying no proof, and is not a theorem.

Write a for the module’s function from natural numbers to integers. The displayed mod denotes the remainder operation: integer remainder for a(n) mod 3, natural-number remainder for n mod 2. The words if, then, else denote Lean’s conditional, with integer-valued branches 1 and 2. A single well-founded recursion constructs s with s(0)=s(1)=0. Define a(k)=1 if k=1 and (3k^2-1)s(k) otherwise; b(m)=1 if m<=1 and 3ms(m) otherwise. For n>=2, s_eq_sum gives s(n)=sum over 1<=k<n of a(k)*b(n-k). No uniqueness or formal identification with the logarithmic generating function is proved here.

Theorem 1.1 (Hanna’s modulo-three conjecture).

$$\forall n:\mathbb{N},(1\le n)⟹((a\left(n\right)\mathrm{mod}3)=\mathrm{if}(n\mathrm{mod}2)=1\mathrm{then}1\mathrm{else}2)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/LogarithmicWeightCatalanParity.hanna_conjecture_a397349_mod_three (✓ std3). ∎

Resolves. Problems/oeis-a397349-logarithmic-weight-mod-three (proved) by D5/S1/Recurrence/Residue/LogarithmicWeightCatalanParity.hanna_conjecture_a397349_mod_three.

Citation. Paul D. Hanna (2026). OEIS A397349, logarithmic-weight coefficients: residue and parity. URL: https://oeis.org/A397349.

Commentary.

For m>=2, b_mod_three uses the explicit factor of three in b(m). In s_eq_sum only k=n-1 survives modulo three, since b(1)=1; the lemma s_mod_three proves s(n) mod 3 = a(n-1) mod 3. The factor 3n^2-1 has remainder 2, so a_mod_three_step gives a(n) mod 3 = (2(a(n-1) mod 3)) mod 3. Induction from a(1)=1 proves the displayed alternation. This argument does not prove the parity conjecture, which remains open.

References

• Truth anchor: D5/S1/Recurrence/Residue/LogarithmicWeightCatalanParity.hanna_conjecture_a397349_mod_three