bibkey: hanna2026a396844 authors: Paul D. Hanna year: 2026 title: OEIS A396844, g.f. satisfying A(xA(x) - 4xA(x)^2) = x^2 doi: null url: https://oeis.org/A396844 claim: “G.f. A(x) satisfies A( xA(x) - 4xA(x)^2 ) = x^2. Conjecture: for n > 8, a(n) == 12 (mod 16) iff n is of the form 2*4^k+1 otherwise a(n) is divisible by 16.” strata_touched:

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

OEIS A396844

Paul D. Hanna’s entry defines the normalized integral generating series by A(x*A(x) - 4*x*A(x)^2) = x^2, with A(x) = x + ..., and states the quoted congruence conjecture. The Lean module constructs this series, proves uniqueness, and proves the corrected classification: for every n >= 2, the residue is 4*(-1)^k modulo 16 when n = 2^k + 1, and zero otherwise. Consequently the conjecture is false: 17 = 2^4 + 1 has residue 4, although 17 is not of the form 2*4^k + 1, where the conjecture requires divisibility by 16. The residue-12 class in the conjecture is unchanged.

Verified locator

• URL: https://oeis.org/A396844