Identifying codes for infinite triangular grids with a finite number of rows

Let $\mathcal{G}_T$ be the infinite triangular grid. For any positive integer $k$, we denote by $T_k$ the subgraph of $\mathcal{G}_T$ induced by the vertex set $\{(x,y)\in\mathbb{Z}\times[k]\}$.A set $C\subset V(G)$ is an {\it identifying code} in a graph $G$ if for all $v\in V(G)$, $N[v]\cap C\neq \emptyset$, and for all $u,v\in V(G)$, $N[u]\cap C\neq N[v]\cap C$, where $N[x]$ denotes the closed neighborhood of $x$ in $G$.The minimum density of an identifying code in $G$ is denoted by $d^*(G)$.In this paper, we prove that $d^*(T_1)=d^*(T_2)=1/2$, $d^*(T_3)=d^*(T_4)=1/3$, $d^*(T_5)=3/10$, $d^*(T_6)=1/3$ and $d^*(T_k)=1/4+1/(4k)$ for every $k\geq 7$ odd. Moreover, we prove that $1/4+1/(4k)\leq d^*(T_k)\leq 1/4+1/(2k)$ for every $k\geq 8$ even.