# 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.

## Key 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 $n_m=m\cdot2^k$