Ulam-Warburton Automaton - Counting Cells with Quadratics

TL;DR

This paper introduces a sequence of quadratic functions that precisely count the total ON cells in the Ulam-Warburton automaton up to generation n, where n has a specific form, providing a mathematical understanding of its growth.

Contribution

It presents a novel quadratic formula for counting ON cells in the Ulam-Warburton automaton based on the generation number's structure.

Findings

Derived quadratic functions for cell counts

Established formulas for specific generation forms

Enhanced understanding of automaton growth patterns

Abstract

This paper is about a sequence of quadratic functions that enumerate the total number of ON cells up to and including generation $n$ of the Ulam-Warburton cellular automaton, where $n$ has the form $n_m=m\cdot2^k$

Figures