bibkey: hanna2026a397241b authors: Paul D. Hanna year: 2026 title: “OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1)” doi: null url: https://oeis.org/A397241 claim: “G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1. Conjecture: a(n) == 2 (mod 3) for n > 2. Conjecture: a(n) == 3 (mod 4) for n > 2.” strata_touched:

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

OEIS A397241

Paul D. Hanna’s entry, dated June 19, 2026, specifies the generating equation and the two coefficient conjectures quoted above. Here [x^n] extracts the degree-n coefficient of a formal power series. The normalized integer sequence has a(0) = a(1) = 1. Its construction, generating equation, uniqueness, and oddness theorem belong to DiagonalPowerRatioAllOdd; the present module uses that same coefficient function and generating series.

Write G = 1/(1-x) for the formal geometric series and B = (1-2x^3)G. The module identifies the generating series with B modulo both three and four. Modulo three, extraction by residue class gives a contracting recurrence for the diagonal residual. Modulo four, the identity B(x)^2 = B(x^2)(1+E), with E = 2(xG+x^6) and E^2 = 2E = 0, gives five polynomial numerator states closed under even and odd coefficient extraction. Strong induction proves the required residuals vanish for every n > 1. The coefficient multiplier in the uniqueness comparison is exactly one, so both reduced equations determine the reductions of the existing integer solution.

Above degree two, B has coefficient -1. Consequently a(n) mod 12 = 11 for n > 2, and reduction gives the two conjectured remainders. The quotient in G is a formal unit inverse, not division of natural numbers.

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• URL: https://oeis.org/A397241