bibkey: hanna2025a380680
authors: Paul D. Hanna
year: 2025
title: "OEIS A380680, bilateral rational-quotient square series"
doi: null
url: https://oeis.org/A380680
claim: "Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare."
strata_touched:
  - D5/S0/Certificates/OeisA380680BilateralModFourRefutation
license: citation-only
triage: anchor

OEIS A380680

The unpaginated entry’s Name is: “G.f. A(x) = Sum_{n=-oo..+oo} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.”

Its Comments state: “The term x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n is invariant under sign reversal of n.”

Formula (2) is: “A(x) = 1 + 2Sum_{n>=1} x^(n^2) * (2 - x^n)^n/(1 - 2x^n)^n.”

The Comments conjecture: “Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.”

The sequence starts at index zero with 1, 4, 6. The formal coefficient definition uses formula (2) over rational power series. At degree m only indices 1 through m are needed: summand n contains X^(n^2). Its coefficient at degree one is four. Since one is a square but four has remainder zero modulo four, the square-residue conjecture is false. The separate A380060 entry defines an implicit series and is not the series considered here.

Verified locator

Source: https://oeis.org/A380680