Rémy Sigrist's Sequence

Overview

Rémy Sigrist's sequence is one of the most visually surprising structures to emerge from simple combinatorial rules. At first glance, the sequence plots resemble mountainous terrain, complete with jagged slopes, basins, and terraces. But behind the landscape lies a strict rule: for any two distinct integers ii and jj, if a(i) = a(j) then a(i) = a(j), then the bitwise AND of ii and jj must be zero. In other words, no two indices that share any binary digits in common are allowed to land on the same value. The result is a finely tuned packing problem over binary representations. The lexicographically earliest solution creates not only a valid sequence but one that yields striking geometric structure when visualized as vertical stacks.

This visual structure is not arbitrary. The zeros in the sequence align exactly with powers of two, while the higher values fill in space under a constraint that mimics binary orthogonality. As the index increases, earlier values fill out a kind of invisible grid, pushing later values higher in a deterministic but unpredictable way. The apparent fractality comes from the recursive logic hidden within the binary expansion of integers. The result is a pattern that feels simultaneously engineered and emergent. This is what happens when discrete logic generates form. A sequence becomes a terrain. A function becomes a topology.

To generate the plot, I implemented the rule directly using the bitwise AND operator. I created a list of counters where each entry tracked the cumulative binary mask used at that level. For each integer ii, I searched for the smallest index aa such that ii shared no bits with the existing value stored at that level. Once I found this position, I updated the mask with ii and assigned that index as the sequence value. This approach guaranteed the constraint was preserved and ensured that every assignment respected the binary independence condition.

I visualized the result by plotting each index against its corresponding sequence value. The image revealed sharp peaks and valleys that traced out recursive forms and suggested deeper structure beneath the surface. I used a binary color palette to emphasize contrast and avoid visual noise. What emerged was a pattern that feels architectural, precise, and unexpected. It is a rare example of visual elegance arising purely from logic. No geometry was added by hand. No randomness was introduced. Only a single condition applied across the integers produced something that resembles natural symmetry and constructed design.

You can find my code repository here.