bibkey: hanna2025a389537
authors: Paul D. Hanna
year: 2025
title: OEIS A389537, g.f. satisfying A(A(x) - x) = A(x)^2
doi: null
url: https://oeis.org/A389537
claim: "G.f. A(x) satisfies A( A(x) - x ) = A(x)^2. Conjecture: for n > 2, a(n) == 2 (mod 4) when n = 3*2^k for k >= 0 otherwise a(n) is divisible by 4."
strata_touched:
  - D5/S1/Recurrence/Invariants/AlternatingDyadicReversionModFour
license: citation-only
triage: anchor

OEIS A389537

Paul D. Hanna’s entry defines the integer generating series by A(A(x) - x) = A(x)^2, with zero constant coefficient and linear coefficient one. It states the coefficient congruence conjecture quoted above. The Lean module constructs this series by compositional reversion, proves its defining equation and uniqueness, and proves both directions of the mod-four classification for every index greater than two.

Verified locator

• URL: https://oeis.org/A389537