k-neighborhood for Cellular Automata
  A neighborhood for d-dimensional cellular automata is introduced that spans the range from von Neumann to Moore neighborhood using a parameter which represents the dimension of hypercubes connecting neighboring cells. The neighborhood is extended to include a concept of radius. The number of neighbors is calculated. For diamond-shaped neighborhoods, a sequence is obtained whose partial sums equal Delannoy numbers. 
Dmitry A. Zaitsev
cs.DM