A PGC (pretty good challenge). Given:

x=(y^2-(n^2-n))/(2n-1)=(y^2-Oblong)/Odd

To prove,

1. If n>0, there will be Integer solution for x iff (2n-1) are the numbers that are divisible only by primes congruent to 1 mod 4 (

http://oeis.org/A004613,

https://oeis.org/A008846,

https://oeis.org/A020882).

2. If n≤0, there will be Integer solution for x iff n=-|2m| negative Even (negative sequence

https://oeis.org/A226485, or twice the negative sequence

http://oeis.org/A094178).