Školjka

%V0 For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$Determine $N(n)$ for all $n\geq 2$. Proposed by Dan Schwarz, Romania