bibkey: hanna2024a378258
authors: Paul D. Hanna
year: 2024
title: "OEIS A378258, a compositional square root and dyadic parity"
doi: null
url: https://oeis.org/A378258
claim: "Conjecture: a(n) is odd iff n is a power of 2."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaA378258SquareRootParity
  - D5/S1/Recurrence/Invariants/HannaA378258ModSix
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A378258

OEIS A378258

The unpaginated Name states: “G.f. satisfies A(x) = B(x) + 5*A(x)^2 where B(A(x) - A(x)^2) = x.”

The Comments state: “Conjecture: a(n) is odd iff n is a power of 2.”

The offset is one. The normalized rational series has constant coefficient zero and linear coefficient one. The formal proposition includes existence and integrality and tests every natural exponent at least one.

The parity assertion is distinct from the entry’s conjecture about residues modulo six and its assertion about the first negative term.

The separate Comments conjecture states: “Conjecture: a(n) is not congruent to 1 nor 5 (mod 6) for n > 1.” Its exact target and bounded literature screen are preregistered in #25601. The corresponding modulo-six source unit reuses the same rational IsGF predicate, quantifies every normalized solution and every natural degree greater than one, and includes existence and an integer coefficient representative. Its reduction modulo three derives reflection symmetry from the equations; symmetry is not an added source assumption. Literature completeness remains ASSUMED-UNVERIFIED; no global priority is asserted.