bibkey: hanna2024a377106
authors: Paul D. Hanna
year: 2024
title: "OEIS A377106: cubic substitution and power-of-two parity"
doi: null
url: https://oeis.org/A377106
claim: "Conjecture: a(n) is odd iff n = 2^k for some k >= 0."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaA377106ParityReversion
license: citation-only
triage: anchor

OEIS A377106: cubic substitution and power-of-two parity

Verified locator

Source: https://oeis.org/A377106

Statement

Conjecture: a(n) is odd iff n = 2^k for some k >= 0.

The entry is unpaginated. Its NAME is:

G.f. A(x) satisfies A(x)^3 = A( x^3 + 9xA(x)^3 ), with A(0)=0, A’(0)=1.

Its COMMENTS give the quoted conjecture. Its offset is one. The encoding uses any rational power series with the three displayed properties and asserts existence, integrality of every positive-index coefficient, and the parity equivalence for the corresponding integer. Reduction modulo two identifies the constructed series with the power-of-two series; uniqueness transfers the coefficient statements to every normalized rational solution.

Literature-screen completeness is ASSUMED-UNVERIFIED. The source entries cite no proof of these conjectures; this is a statement about the checked entries, not a claim of exhaustive literature coverage.