Start with two numbers. Then forever add the smallest number that keeps you clear of any evenly-spaced trio. The starting pair is the only choice you ever make — and it decides whether what grows is a crystal or a cloud.
Three numbers are in arithmetic progression when the gap between
the first two equals the gap between the last two: 1, 4, 7, or 0, 5, 10.
The rule of this whole page is a single prohibition — never let three of your
numbers fall into such a trio — and a single appetite: at each step, take the
smallest number that obeys.
That is the entire machine. It is blind and it is greedy. It looks only for the next legal brick and lays it. Roman Stanley and Andrew Odlyzko set it running in a 1978 Bell Labs memo, and it has been quietly refusing to be understood ever since. Because the machine does something strange: from some starting pairs it builds an object of perfect, provable order — and from others, starting pairs barely different, it builds something that looks like noise and that, nearly fifty years on, no one can classify.
Run it yourself. Pick a seed and watch the rule reject the numbers that would betray it.
Two things are worth noticing already. First, the rule only ever has to look
backward: a new number c is illegal exactly when some earlier number
b sits halfway between an even-earlier number and c — that is, when
a + c = 2b for two numbers a<b you have already kept. Second, the
sequence that starts 0, 1 has a secret.
Look at the seed 0, 1 in base three. The terms are
0, 1, 10, 11, 100, 101, 110, 111, 1000… — and there is the pattern in plain
sight: these are exactly the whole numbers whose base-three writing never uses the
digit 2. The greedy machine, choosing blindly, has reconstructed the integer
cousin of the Cantor set — the
numbers you keep when you forbid the middle third, over and over.
It is not a coincidence and it is not approximate. There is a closed form:
the n-th term is what you get by writing n in binary and then
reading those same digits as base three. Term number 5 is binary 101,
read in base three as 1·9 + 0·3 + 1 = 10 — and the tenth thing the machine
builds is indeed 10. The instrument below checks the greedy rule against that formula for
a thousand terms, live, in your browser.
| n | n in binary | read in base 3 | aₙ (decimal) | aₙ base 3 |
|---|
So one seed gives a crystal: self-similar, periodic in its base-three digits, growing
like n1.585 (that exponent is log₂3, the dimension
fingerprint of the thirds-removing rule). Change the seed to 0, 4 and the
same blind machine produces this:
No base hides in it. No formula is known for it. Its terms thin out far faster than the crystal's — and whether they obey any clean growth law is an open question. This is the cloud.
Odlyzko and Stanley conjectured in 1978 that every one of these greedy sequences eventually falls into one of two kinds:
Locally periodic in base three; grows like Θ(nlog₂3) ≈
n1.585. For seeds like 0,1 and 0,2 the structure can be
proven exactly. Rolnick (2017) gave a clean definition of "regular" — a purely
local repeating pattern — and proved that regularity forces precisely this
asymptotic growth. The crystal above is the cleanest member.
Seemingly random; conjectured to grow like Θ(n² / log n). But here the honesty matters: no Stanley sequence has ever been proven to be Type 2. The "cloud" seeds are only observed to be chaotic — we have never shown that any of them fails to be a crystal in disguise, nor that it obeys the n²/log n law. In December 2025 Nat Sothanaphan went further and gave numerical evidence that the conjectured rate may be wrong — that the cloud might in fact grow faster, like n2−δ. That work is explicitly heuristic, not a proof. The cloud is genuinely unsettled ground.
You can watch the fork happen. The instrument plots the actual terms of each sequence
on a log–log scale, where a pure power law aₙ ~ np shows up as a
straight line of slope p. The crystal locks onto slope log₂3; the
clouds climb visibly steeper, drifting toward the unreachable slope 2. The slopes
quoted are measured live from the data — finite-window fits, not claimed limits.
It would be easy to oversell a fifty-year-old open problem, so here is the ledger, kept
strictly. Proven: the crystal S(0,1) is the no-digit-2 numbers,
with the binary-read-as-ternary closed form (you just watched the check pass). Rolnick's
regularity theorem pins the growth of the structured seeds to Θ(nlog₂3).
And Moy (2011) proved a law that binds every Stanley sequence whatsoever: its count
of terms below x is at least about √2·√x — equivalently, no Stanley
sequence can grow faster than roughly n²/2. That is the ceiling the dashed
slope-2 line marks, and the clouds press against it without touching.
Not proven — the live frontier: that the dichotomy is real (maybe there
are sequences that are neither crystal nor cloud); that any specific seed is a cloud rather
than a crystal we haven't decoded; and that the clouds grow like n²/log n at all
— a claim now under active challenge. The machine is trivial to state and a child could run
it by hand. What it builds, we mostly cannot yet explain. That gap is the whole reason this
page exists: the verification venue does not only re-derive settled truths, it points the
same honest check at the edge of the map and reports, exactly, how far the light reaches.
Every number on this page is computed in your browser
from the greedy rule, and reproduced offline by research/stanley-sequences/verify.mjs
in the project repository (13/13 checks: A005836 match, the closed form, 3-freeness by
independent brute force, the measured slopes, and Moy's bound on the data). Nothing here is
asserted that the verifier does not recompute.