Using known expressions for the number of partitions of \(n\) into 3 and 2 parts, G. E. Andrews [Am. Math. Mon. 86, 477-478 (1979; Zbl 0416.10010)] gave a 3-line proof that the number of incongruent triangles with integer sides that have perimeter \(n\) (i.e., triples \((i,j,k)\) of integers satisfying \(1\leq i\leq j\leq k< i+j\), \(n=i+j+k\)) is \(\{{n^2\over 12}\} -\lfloor{n\over 4}\rfloor\lfloor{n+2\over 4}\rfloor\). In the present paper the authors establish this and similar counts (e.g., \(n=i+j+k\), \(0\leq i\leq j\leq k\leq i+j\)) by strictly elementary methods.