bibkey: hanna2024a374571
authors: Paul D. Hanna
year: 2024
title: "OEIS A374571, g.f. satisfying A(x) = A(x^2) - x*A(x^2)^2"
doi: null
url: https://oeis.org/A374571
claim: "Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2. Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's."
strata_touched:
  - D5/S1/Recurrence/Parity/FibbinarySquareSubstitutionParity
license: citation-only
triage: anchor

OEIS A374571

Paul D. Hanna’s entry of July 11, 2024 defines the sequence by the displayed functional equation and states the Fibbinary parity conjecture. The normalization is A(0)=1. The equation alone leaves the constant coefficient free: its degree-zero identity is A(0)=A(0).

The Lean module constructs the integer coefficients, proves the generating equation and uniqueness with constant coefficient one, and proves the conjecture for every positive index. Its predicate Fibbinary n is n &&& (n >>> 1) = 0. At bit position i, this intersection is the conjunction of bit i and bit i+1 of n, so its vanishing means no adjacent one bits. The parity argument also covers index zero internally.

Verified locator

• URL: https://oeis.org/A374571
• Fibbinary sequence: https://oeis.org/A003714