bibkey: hanna2026a379764
authors: Paul D. Hanna
year: 2025
title: "OEIS A379764, theta-four bilateral generating function and coefficient congruences"
doi: null
url: https://oeis.org/A379764
claim: "Conjecture: for n > 0, a(n) is odd iff n is a square; and a(n) == (-1)^(n-1) (mod 4) iff n is a square, otherwise a(n) is divisible by 4 if n is nonsquare."
strata_touched:
  - D5/S1/Recurrence/Bilateral/HannaThetaFourModFour
license: citation-only
triage: anchor

OEIS A379764

Paul D. Hanna’s OEIS A379764 gives the bilateral generating-function equation

G.f. A(x) satisfies Sum_{n=-oo..+oo} (4^nA(x)^n - 4A(x))^(n+1) = 2theta_4(x) = 2Sum_{n=-oo..+oo} (-x)^(n^2).

with offset 1 and a(1)=1. The entry records two conjectures:

Conjecture: for n > 0, a(n) is odd iff n is a square.

Conjecture: for n > 0, a(n) == (-1)^(n-1) (mod 4) iff n is a square, otherwise a(n) is divisible by 4 if n is nonsquare.

The formal statement uses coefficientwise symmetric windows for the bilateral sum, the rational Laurent-series embedding, and the normalization constantCoeff A = 0, coeff 1 A = 1.

Verified locator

DOI: null

Source: https://oeis.org/A379764