bibkey: hanna2019a324300
authors: Paul D. Hanna
year: 2019
title: "OEIS A324300: Gaussian generating function and square parity"
doi: null
url: https://oeis.org/A324300
claim: "Odd terms occur only at n = k*(k+2) for k >= 0 (conjecture); a(n*(n+2)) = 1 (mod 2) for n >= 0."
strata_touched:
  - D5/S1/Recurrence/Algebraic/HannaGaussianShiftedSquareParity
license: citation-only
triage: anchor

OEIS A324300

Verified locator

Source: https://oeis.org/A324300

The unpaginated native text states:

%I A324300 #14 Feb 23 2019 11:53:31
%N A324300 G.f.: Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
%C A324300 It is remarkable that the generating function results in a power series in x with only real coefficients.
%C A324300 Odd terms occur only at n = k*(k+2) for k >= 0 (conjecture);
%F A324300 G.f. A(x) is defined by the following series.
%F A324300 (1) Sum_{n>=0} x^n * (x^(n+1) + i)^n / (1 + i*x^(n+1))^(n+1).
%F A324300 a(n*(n+2)) = 1 (mod 2) for n >= 0.
%e A324300 A(x) = 1/(1+i*x) + x*(x^2+i)/(1+i*x^2)^2 + x^2*(x^3+i)^2/(1+i*x^3)^3 + x^3*(x^4+i)^3/(1+i*x^4)^4 + x^4*(x^5+i)^4/(1+i*x^5)^5 + x^5*(x^6+i)^5/(1+i*x^6)^6 + x^6*(x^7+i)^6/(1+i*x^7)^7 + x^7*(x^8+i)^7/(1+i*x^8)^8 + ...
%e A324300 Note that the imaginary components in the above sums vanish.
%o A324300 (PARI) {a(n) = my(SUM = sum(m=0, n, x^m*(x^(m+1) + I +x*O(x^n))^m / (1 + I*x^(m+1) +x*O(x^n))^(m+1) ) ); polcoeff(SUM, n)}
%O A324300 0,3
%A A324300 _Paul D. Hanna_, Feb 21 2019

The rational summands are interpreted as formal power series over the complex numbers. Each positive-degree coefficient is a finite sum. The leading factor supplies local finiteness without a constant replacement. The assertion includes an ordinary integer representative of each coefficient, with oddness equivalent to N = k*(k+2) for a natural index.

Literature-screen completeness is ASSUMED-UNVERIFIED. Issue #21418 records the source revision and the unread dissertation and citation-screen gaps.

Exact-arithmetic fidelity evidence: the-omega-institute/trureturing-experiments, commit e0de2857d1d1616ee67af65221cc0028b888e4dd, docs/reports/oeis-a323689-hanna-square-parity/check.py, SHA256 b1a69c4f5dc4789d36755d31f212795fcb209457a7e947e60f8b1f8bf94305a4. The preregistration records exit 0 and ALL_CHECKS_OK; finite computations do not establish the unbounded assertion.