slug: oeis-a380558-parity bibkey: hanna2025a380558 doi: null url: https://oeis.org/A380558 triage: theorem motivation_gids:

• D5/S1/Recurrence/Parity/A380558

Parity of the A380558 square-reversion series

Problem

OEIS A380558, by Paul D. Hanna (February 13, 2025), defines the generating series A(x) = Sum_{n>=2} a(n)*x^n through A(x - A(x)) = x^2/(1 - x^2). Its comment states:

Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.

The formal target constructs an integral series with zero coefficients at degrees zero and one, proves that source equation and uniqueness among all similarly normalized integral solutions, and proves for every natural n:

Odd ([x^n]A) <-> n = 2 or there exist m,j in N with n = 2*m and 3*2^j <= m < 4*2^j.

This is the source conjecture with its natural zero extension at degrees zero and one. A004760 has offset one: its first values are 0,1,3,6,7,.... The condition k>1 retains 1, giving n=2; all later values lie in the half-open dyadic intervals displayed above. Neither m nor j is the OEIS index k.

Motivation

This was a named, literature-open parity conjecture in a 2025 OEIS entry. The target is its complete unbounded bidirectional assertion, with the source series identified uniquely over the integers.

Gap

The A380558 entry’s formula field already supplies A = B^2/(1-B^2) with B(x-A(x))=x. The project already proves the integral reversion series B, its uniqueness, and its reduction to a characteristic-two lacunary series L in D5/S1/Recurrence/Parity/QuadraticSquareReversionDyadicSupportParity. The neighboring invariant module proves L = X + X^2 + X^2 L^2. These are premises, not new claims of this resolution.

The A380558 entry still labels parity a conjecture in revision 12 on 2026-09-28. The project PR search for A380558 returned no prior PR; the pre-proof literature and code search, including its stated access limits, is recorded in preregistration issue #8272. This is a not-found-in-searched-scope assessment, not a claim of exhaustive priority. The literature note records the exact OEIS revision fields and retrieval hashes for both A380558 and A004760.

Route

The new characteristic-two calculation sets T=L/(1-L) and derives T=X+X^2+(1+X)T^2. Frobenius then gives t_0=0, t_1=1, t_2=0, and t_n=t_floor(n/2) for every n>=3. Strong induction identifies the ones as n=1 or a member of [3*2^j,4*2^j). The integral source series reduces to T^2, excluding all odd degrees and doubling the interval indices.

For source identity, substitution carries the existing reversion equation from B to A=B^2/(1-B^2). For uniqueness, the compositional inverse of X-F converts any normalized source solution F back to a series satisfying the existing unique B equation. The inverse-unit denominators have constant coefficient one throughout.

Falsifier

An index where the unique normalized source series has the opposite parity from the stated support would falsify the conjecture. A finite prefix cannot establish the universal result; a source-normalization or formal-division error would invalidate the identification with OEIS rather than refute it.

Evidence

• Lean source: D5/S1/Recurrence/Parity/A380558.lean, public theorem result.
• Source-bound registration: Reg/D5/S1/Recurrence/Parity/A380558.lean; the current Lean report records declared_validated with a complete source-equivalence bridge.
• The theorem and registration each depend only on propext, Classical.choice, and Quot.sound in the current report.
• The literature note records the exact OEIS fields, revision markers, retrieval hashes, and plain-text locators.

Triage

theorem. The result proves the conjecture for every natural degree of the unique normalized integral solution.

ASSUMED-UNVERIFIED

The literature search is bounded and does not certify first-publication priority. Source-to-Lean identification is reviewed against the OEIS fields, but is not a kernel theorem. This result does not settle the separate A185897 conjecture or claim novelty for the published reversion transform.

Preregistration: issue #8272.