Multiple Lattice Packings and Coverings of the Plane with Triangles

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 $(2j+1){\prod }_{p\mathrm{\mid }2j+1}(1-\frac{2}{p})$, where the product is over the distinct prime numbers dividing $2j+1$.