bibkey: hanna2024a368628 authors: Paul D. Hanna; OEIS Foundation Inc. year: 2024 title: A368628 — a piecewise square and fourth-power generating function doi: null url: https://oeis.org/A368628 claim: The source defines the generating function and conjectures oddness exactly at base-four repunit indices; the proof is derived in this repository. strata_touched:

• D5/S1/Recurrence/PiecewiseConvolutionPowersOfFour license: citation-only triage: anchor

A368628: source equation and parity conjecture

Verified locator

url: https://oeis.org/A368628

doi: null

The official internal page https://oeis.org/A368628/internal was read on 2026-09-10 (HTTP 200). The entry credits Paul D. Hanna, January 10, 2024. Its NAME and formula (1) define

A(x) = 1 + x(A(x)^2 - A(-x)^2)/2 + x(A(x)^4 + A(-x)^4)/2.

The COMMENT says: “Conjecture: a(n) is odd when n = (4^k - 1)/3 for k >= 0, and even elsewhere.” The entry’s first ten terms are 1, 1, 2, 14, 32, 345, 810, 10492, 24880, 356252.

Coefficient interpretation

The constant coefficient is 1. For n>0 even, the coefficient is degree n-1 of A squared; for n odd, it is degree n-1 of A to the fourth power. Both are finite natural-number convolutions using only earlier coefficients. This determines the sequence independently of the parity conjecture. The Lean definition uses well-founded recursion and a zero extension of the earlier coefficients; the coefficient recurrence proves that the extension does not affect the result.

Literature search boundary

The four directly referenced entries A368593, A368626, A368627, A368629 were read on the same date (HTTP 200, NAME/COMMENT/FORMULA/LINK/CROSSREF fields). None supplies this parity proof. A368628 lists Robert Israel’s July 2, 2026 algebraic polynomial and order-68 recurrence; these are not claimed here as parity proofs. The linked recurrence was downloaded, but its polynomial coefficients were not individually verified (ASSUMED-UNVERIFIED).

Authenticated GitHub code search for A368628 with language Lean returned 0 results; arXiv API all:A368628 returned 0 results. These are bounded searches, not a claim of exhaustive novelty. The parity proof and its coefficient reductions are repo-derived; this note attributes the source definition and conjecture only. No DOI is supplied by the OEIS entry.