I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ such that
$$ n^2 = a^2 + b^2+b.$$
For example, $$9 = 1^2 + 8 = 2^2 + 5,$$ so it is such a number; however, $16=2^2 + (2 \cdot 6)$, so it is not.
This paper http://math.nju.edu.cn/~zwsun/111o.pdf claims a proof about numbers which cannot be written as the sum of a square and two [not necessarily equal] triangular numbers — I will try to adapt their proof if I can't find another.
Any references or hints on how to approach the problem would be appreciated.