bibkey: hanna2026a396846 authors: Paul D. Hanna year: 2026 title: “OEIS A396846, l.g.f. log(1 + x + Sum 4n/(4n^2-1) a(n) x^n), mod-8 conjecture” doi: null url: https://oeis.org/A396846 claim: “L.g.f. Sum_{n>=1} a(n)x^n/n = log(1+x + Sum_{n>=2} 4n/(4*n^2-1) * a(n)*x^n ). Conjecture: a(n) == [1,7,5,7] repeating (mod 8) for n >= 1.” strata_touched:

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

OEIS A396846

Paul D. Hanna’s entry, dated June 24, 2026, specifies the logarithmic generating function and the second conjecture quoted above. The sequence starts at index one with 1, 15, 805, 75663, 10440441, 1954144335. The separate conjecture modulo three is not asserted here as a theorem, and it does not need to be: it already follows from frozen content filed under a different entry. Modulo three the factor 4n^2 - 1 vanishes whenever 3 does not divide n, so theta F = X and theta^2 F = X for F = Sum a(n) X^n; differentiating theta H = F H twice more gives theta^3 H = (X + F^3) H, and theta^3 = theta in characteristic three, so cancelling the unit H leaves F = X + F^3. The frozen D5/S3/Arith/TernaryTraceSupport carries a series t with t^3 = t - X and zero constant coefficient, whose coefficients are one exactly at the powers of three. For any solution f of f = X + f^3, put q = 1 - (f^2 + f t + t^2); then q (f - t) = 0 and q has constant coefficient one, hence is a unit, so f = t. No induction is required, and the residue is one rather than two at every supported index.

Writing that chain out as a new Lean declaration would be a second source for a truth the graph already carries, so it is recorded here instead.

The formal meaning of the logarithm is B * H = X * derivative H, where B = Sum_{n>=1} a(n) X^n, [X^0]H = [X^1]H = 1, and [X^n]H = 4n a(n)/(4n^2-1) for n >= 2. Thus the series with coefficients a(n)/n and zero constant coefficient is the formal logarithm of H.

The module constructs integral normalized coefficients c(n) with a(n) = (4n^2-1)c(n) for n >= 2. Their recurrence proves the displayed formal equation and its rational coefficient shape. A coefficient induction proves uniqueness even among rational solutions with the same initial terms. Oddness reduces the normalized convolution modulo eight to four times the sum of its indices. The latter is four-periodic modulo eight, giving residues 1, 7, 5, 7 for successive positive indices.

Verified locator

• URL: https://oeis.org/A396846