k-neighborhood for Cellular Automata

TL;DR

This paper introduces a flexible k-neighborhood for d-dimensional cellular automata, unifying various neighborhood types and extending to include a radius, with mathematical analysis of neighbor counts and connections to Delannoy numbers.

Contribution

It presents a novel, parameterized neighborhood model for cellular automata that generalizes existing neighborhood types and incorporates a radius concept, with mathematical properties analyzed.

Findings

Derived formulas for neighbor counts in the new neighborhood model

Connected neighborhood structures to Delannoy numbers

Unified framework encompassing von Neumann and Moore neighborhoods

Abstract

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.