Stochastic Forestfire Expansion

Overview

The forest fire sequence is defined by a deceptively simple rule: each term is the smallest positive integer that avoids forming any arithmetic progression of length three with earlier terms. That is, for all j and k, the condition a(j), a(j+k), a(j+2k) forming an arithmetic progression must be avoided. Despite the simplicity of this constraint, the sequence produces a surprisingly rich and unpredictable structure, full of sudden jumps, plateaus, and repetitions. It reflects a delicate balance between regularity and disruption, where the avoidance of symmetry drives complexity.

The plot evokes the appearance of smoke drifting across a burning field—an emergent shape born from discrete arithmetic rules. Dense regions suggest areas of repetitive small values, while abrupt scatterings of higher values punctuate the landscape like bursts of energy. The cumulative effect of these tiny decisions not to form a simple progression is astonishing: the result is a turbulent, organic form that seems to pulse with chaotic motion. This visual resemblance to natural phenomena like smoke or fire inspired the nickname "forest fire" for the sequence.

Mathematically, the key idea is to reject any candidate value that completes a three-term arithmetic sequence when substituted at position n. In code, for each n, we maintain a set of forbidden values constructed using the formula 2a(n−k)−a(n−2k) across all k such that n−2k ≥ 0. The first integer not found in this set becomes a(n). While the rule is short, it encodes a dense network of dependencies between terms that grow increasingly entangled. This makes the sequence fascinating both as a recursive object and as a dynamic pattern generator in Python-based visualizations.

You can find my code repository here.