bibkey: hanna2026a396793 authors: Paul D. Hanna year: 2026 title: “OEIS A396793, g.f. satisfying A(x) A(A(x)) = x^2 + 9 x^3” doi: null url: https://oeis.org/A396793 claim: “G.f. A(x) satisfies A(x) * A(A(x)) = x^2 + 9*x^3. Conjecture: a(n) == 0 (mod 3) for n > 1.” strata_touched:

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

OEIS A396793

Paul D. Hanna’s entry, dated June 7, 2026, specifies the generating equation and coefficient conjecture quoted above. The normalization is A(0) = 0 and a(1) = 1, with a(n) the coefficient of x^n in A. The listed terms, starting at index 1, are 1, 3, -12, 81, -645, 5367.

The formal module constructs the unique normalized integer power series satisfying the equation. For an integer series F congruent to X modulo 3 and with zero constant coefficient, its product F(X) F(F(X)) is congruent to X^2 modulo 9. This lifting identity makes each coefficient correction an exact integer quotient by 3 and a multiple of 3. Comparing a solution with its truncated prefix then proves the conjecture for every n > 1.

Verified locator

• URL: https://oeis.org/A396793