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orl wrote: Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$ So $a \le b \le c \le n$ and obviously $a+b>c$. Let $P_n(k)$ denotes the number of such triplets with $c=k,k=1,2,.....,n$. For $k$ even, $P_n(k)=k+(k-2)+(k-4)+...+4+2=\frac{k^2+2k}{4}$, and for $k$ odd, $P_n(k)=\frac{(k+1)^2}{4}$ So $P_n(1)+P_n(2)+.....+P_n(n)=\frac{n(n+2)(2n+5)}{24}$ when $n$ is even, and $\frac{(n+1)(n+3)(2n+1)}{24}$ when $n$ is odd.