Counting with Hiccup Sequences
A simple counting rule reveals hidden patterns in sequences of numbers
A hiccup sequence follows a straightforward rule: start at a number and keep adding 1 or 2 depending on whether you've seen your current position before. Mathematicians proved that this simple rule produces remarkably orderly patterns—the missing numbers always have gaps that follow a predictable sequence, and these gaps connect to well-studied shapes in mathematics like polygonal numbers.
This work shows how simple iterative rules can generate hidden structure. The patterns discovered here appear across multiple areas of mathematics—from counting polyominoes (puzzle pieces) to Ramsey theory (how unavoidable patterns emerge in large systems)—suggesting that hiccup sequences may be a fundamental tool for understanding when and why order emerges from simple repetition.