ISL 1989 Problem 49

Let $t(n)$ for $n = 3, 4, 5, \ldots,$ represent the number of distinct, incongruent, integer-sided triangles whose perimeter is $n;$ e.g., $t(3) = 1.$ Prove that

$$t(2n-1) - t(2n) = \left[ \frac{6}{n} \right] \text{ or } \left[ \frac{6}{n} + 1 \right].$$