Example. The number of integer lattice points that satisfy the conditions $$-n\leq x,y,z\leq n,\quad -s\leq x+y+z\leq s$$ for some $n$, $s\in\mathbb N$, is equal to $$\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\left(\frac{\sin \frac{2n+1}{2}t}{\sin\frac {t}{2}}\right)^3\frac{\sin \frac{2s+1}{2}t}{\sin \frac{t}{2}} dt.$$