Given a convex disk $K$ and a positive integer $j$, let $\delta_L^j(K)$ and $\vartheta_L^j(K)$ denote the $j$-fold lattice packing density and the $j$-fold lattice covering density of $K$, respectively. I will prove that for every triangle $T$ we have that $\delta_L^j(T)=\frac{ 2j^2}{ 2j+1}$ and $\vartheta_L^j(T)=\frac{ 2j+1}{ 2}$. Furthermore, I also obtain that the numbers of lattices which attain these densities both are $(2j+1)\prod_{ p|2j+1}(1-\frac{ 2}{ p})$, where the product is over the distinct prime numbers dividing $2j+1$.
Categories: math.MG math.NT