Bijection Difficult

Problem - 4424

Let $S_n$ be the number of non-congruent triangles whose sides' lengths are all integers and circumferences equals $n$. Show that $$S_{2n-1}-S_{2n} = \left\lfloor\frac{n}{6}\right\rfloor\quad\text{or}\quad\left\lfloor\frac{n}{6}\right\rfloor +1$$

where $\lfloor{x}\rfloor$ returns the largest integer not exceeding the real number $x$.


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