# $n$ incongruent triangles with integer sides and perimeter $n$

What is the only positive integer $$n$$ such that there are exactly $$n$$ incongruent triangles with integer sides and perimeter $$n$$?

I have found that the answer to the above problem is $$n = 48$$. I know how this problem relates to Alcuin's sequence: https://oeis.org/A005044.

How can I show that $$n = 48$$ is the $$\textbf{only}$$ solution? I have been able to show that there are no solutions for $$n < 48$$.

At the linked OEIS site there is the formula $$a_{2n-3}=a_{2n}={\rm round}\left({n^2\over12}\right)\ ,$$ hopefully valid for all $$n\geq2$$. This should squeeze $$n$$ sufficiently to show that there are no $$j\ne48$$ with $$a_j=j$$.