On number of nonzero cells in some two-dimensional reversible second-order cellular automata

TL;DR

This paper derives recursive formulas for counting nonzero cells over time in specific two-dimensional reversible second-order cellular automata, starting from a single active cell.

Contribution

It introduces recursive equations for the evolution of nonzero cells in certain reversible cellular automata, a novel analytical approach for these systems.

Findings

Recursive formulas for nonzero cell counts are established.

Initial configuration with a single active cell is analyzed.

Results provide insights into the automata's dynamics.

Abstract

Recursive equations for the number of cells with nonzero values at $n$-th step for some two-dimensional reversible second-order cellular automata are proved in this work. Initial configuration is a single cell with the value one and all others zero.