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."
that may affect the evolution of a two-dimensional cellular automaton on a square grid. The von Neumann neighborhood of range 
is defined by 
, 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). 