Given positive integers $n$ and $k \leq n$. Consider an equilateral triangular board with side $n$, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles, $\ldots$ , in the bottom row there are $n$ circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are $3n$ such lines) contains no more than $k$ checkers. Denote by $T(k, n)$ the largest possible number of checkers in such a placement. a) Prove that the following upper bound is true: $$T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor$$b) Find $T(1,n)$ and $T(2,n)$ D. Zmiaikou