bibkey: hanna2025a389472
authors: Paul D. Hanna
year: 2025
title: "OEIS A389472, g.f. satisfying A(x) = A(x^2 + x^3)/x^2 - 1"
doi: null
url: https://oeis.org/A389472
claim: "G.f. satisfies A(x) = A(x^2 + x^3)/x^2 - 1. Conjecture: a(2*n-1) = 0 (mod 2) for n > 1. The separate conjecture modulo 3 is not claimed here."
strata_touched:
  - D5/S1/Recurrence/Parity/SquareCubeSubstitutionOddIndexParity
license: citation-only
triage: anchor

OEIS A389472

Paul D. Hanna’s entry, dated October 15, 2025, defines the sequence by A(x) = A(x^2 + x^3)/x^2 - 1 and states: “Conjecture: a(2n-1) = 0 (mod 2) for n > 1.” The entry also states “Conjecture: a(3n-1) = 0 (mod 3) for n > 1.” Only the first conjecture is addressed here.

The formal normalization is a(0)=0, a(1)=a(2)=1, matching the entry’s initial terms and PARI program. The cross-multiplied identity is x^2 (A(x)+1) = A(x^2+x^3). Its degree-two coefficient forces a(1)=1, while degree three imposes no condition on a(2). The seed a(2)=1 therefore selects the intended solution. The Lean module constructs this integer series, proves its functional equation and uniqueness with all three seeds, and proves the first conjecture by binomial parity descent.

Verified locator

• URL: https://oeis.org/A389472