A diamond-shaped neighborhood that can be used to define a set of cells surrounding a given cell that may affect the evolution of a two-dimensional
cellular automaton on a square
grid. The von Neumann neighborhood of range is defined by

von Neumann neighborhoods for ranges , 1, 2, and 3 are illustrated above. The number of cells
in the von Neumann neighborhood of range is the centered square
number ,
the first few of which are 1, 5, 13, 25, 41, 61, ... (OEIS A001844).

Gray, L. "A Mathematician Looks at Wolfram's New Kind of Science." Not. Amer. Math. Soc.50, 200-211, 2003.Sloane,
N. J. A. Sequence A001844/M3826
in "The On-Line Encyclopedia of Integer Sequences."