Moore Neighborhood

A square-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 Moore neighborhood of range  is defined by

Moore neighborhoods for ranges , 1, 2, and 3 are illustrated above. The number of cells in the Moore neighborhood of range  is the odd squares , the first few of which are 1, 9, 25, 49, 81, ... (OEIS A016754).

REFERENCES:

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 A016754 in "The On-Line Encyclopedia of Integer Sequences."