bibkey: hanna2025a389536
authors: Paul D. Hanna
year: 2025
title: "OEIS A389536, G.f. A(x) = A(x^2 + 2*x^3)/x"
doi: null
url: https://oeis.org/A389536
claim: "G.f. A(x) satisfies: A(x) = A(x^2 + 2*x^3) / x. Conjecture: for n > 1, a(n) == 2 (mod 4) when n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4."
strata_touched:
  - D5/S1/Recurrence/Invariants/SubstitutionSquareCubeModFour
license: citation-only
triage: anchor

OEIS A389536

Paul D. Hanna’s entry, dated October 28, 2025, defines the generating function by A(x) = A(x^2 + 2*x^3) / x and states the quoted coefficient conjecture. The normalization is a(1) = 1; comparing the coefficient of x^2 in x*A(x) = A(x^2 + 2*x^3) gives the tautology a(1) = a(1), so this normalization is free. The Lean module constructs the integer series with zero constant term, proves the normalized equation and uniqueness, and proves the conjecture for every index greater than one.

Verified locator

• URL: https://oeis.org/A389536