On an equidistant one-dimensional lattice of integers $i$ the
discrete Laplacian may be represented by the one-step or two-step
second-order difference operators $\triangle(1)$ or
$\triangle(2)$, respectively. In such a setting the equation
$\triangle(1)\phi=\triangle(1)Z$ possesses a `local' solution
$\phi(i)=Z(i+1)+2Z(i)+Z(i-1)$. The author describes a
generalization of this observation to two dimensions (honeycomb
lattice, with the doubling of distances replaced by the doubling
of angles) and to three dimensions (a body-centered tetrahedral
lattice, with the doubling of angles replaced by the doubling of
spherical angles).



MR2172003 Zakrzewski, Wojtek J. Laplacians on lattices. J.
Nonlinear Math. Phys. 12 (2005), no. 4, 530--538.