February 8, 2026
It’s numbers all the way down
Kolakoski Sequence
The self-describing number spiral that’s breaking brains and sparking nerd wars
TLDR: The Kolakoski sequence is an infinite pattern of 1s and 2s that describes itself, with open questions like whether 1s and 2s appear equally. Comments swing from helpful explainers to coder flexes and jokes about it being #2, fueling a debate over deep math vs clever gimmick.
Meet the Kolakoski sequence: an infinite stream of 1s and 2s that literally describes itself. Think numbers writing their own autobiography—each digit tells how long the next streak should be. A hypnotic spiral visualization sent the comment section into full-on math meltdown: half the crowd is mesmerized, the other half is clutching their temples.
Strong reactions came fast. Confused readers rallied around MontyCarloHall’s chalkboard-style breakdown, calling it the moment “the penny dropped.” Coders showed up flexing, with nwellnhof linking a slick bit-fiddling generator that sparked a mini flame war: elegant wizardry versus cursed hacks. Meanwhile, jokers couldn’t resist that it’s entry #2 in the Online Encyclopedia of Integer Sequences—“of course it’s 2,” cue the numerology memes.
Then the drama escalated: math purists debated whether 1s and 2s show up equally often (it’s still unproven), while pedants argued if calling it a “fractal” is poetic or misleading. Hot takes split the room: some call it profound self-reference, others dismiss it as “run-length encoding with eyeliner.” Through it all, the meme chorus keeps yelling it’s numbers all the way down, as everyone tries not to get hypnotized by that spiral.
Key Points
- •The Kolakoski sequence is an infinite {1,2} sequence equal to its own run-length encoding.
- •Its construction is self-referential and reversible: terms generate runs, and runs generate terms.
- •It is not eventually periodic and is cube-free, with several recurrence properties unresolved.
- •A conjecture states the density of 1s is 1/2; proven upper bounds are 0.50084 (Chvátal) and 0.500080 (Nilsson).
- •Bertran Steinsky provided a recursive formula for computing the i-th term; the sequence is cataloged as OEIS A000002.