bibkey: hanna2026a393173
authors: Paul D. Hanna
year: 2026
title: OEIS A393173, squared-index inverse-diagonal parity
doi: null
url: https://oeis.org/A393173
claim: 'G.f. A(x) satisfies [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1. Conjecture: for n > 0, a(n) is odd iff n is a power of 2.'
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaA393173PowerTwoParity
license: citation-only
triage: anchor

OEIS A393173

Verified locator

Source: https://oeis.org/A393173

The unpaginated entry’s Name reads:

G.f. A(x) satisfies [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1.

Its Comments state:

Conjecture: for n > 0, a(n) is odd iff n is a power of 2.

The offset-zero data fix a(0)=1. The formal statement quantifies over all normalized rational power series satisfying this equation and asserts existence, integrality of every positive-degree coefficient, and the quoted parity equivalence. The separate modulo-four conjecture is outside that statement.

The coefficient recursion is a strict contraction on normalized series. Reduction modulo two identifies its fixed point with one plus the power-of-two series. The characteristic-two identity S+S²=X makes the even and odd inverse-diagonal conditions vanish separately.

The source is cited under the OEIS End-User License Agreement: https://oeis.org/LICENSE. No program text is reproduced.