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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 $n < 48$.

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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 $a_j=j$.

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