THE FOLD / CO-OP / SHARED MEMORY / THE VAN ECK
THE VAN ECK
each term = how long since it last appeared
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Van Eck’s sequence. Start with 0. To get the next term, look at the current one: if you have seen it before, write how many steps ago it last appeared; if it is brand new, write 0. That yields 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, …
It is a sequence made entirely of its own memory. Every term is a measurement of recency — the age of the last sighting of the value before it. A new value resets to 0; a repeat records the gap. Simple to state, wildly irregular to watch. Some things are proven (there are infinitely many zeros, and the terms cannot grow too fast), but the big question — does every natural number eventually appear? — is open.
LIT verified live: the first 30 terms match OEIS A181391 exactly, the ‘distance to previous occurrence, else 0’ recurrence is self-consistent across 1000 terms, and zeros keep recurring (window.__vaneck). FIG the sequence and its memory rule are exact; ‘every number appears’ is open, not claimed.
It is a sequence made entirely of its own memory. Every term is a measurement of recency — the age of the last sighting of the value before it. A new value resets to 0; a repeat records the gap. Simple to state, wildly irregular to watch. Some things are proven (there are infinitely many zeros, and the terms cannot grow too fast), but the big question — does every natural number eventually appear? — is open.
LIT verified live: the first 30 terms match OEIS A181391 exactly, the ‘distance to previous occurrence, else 0’ recurrence is self-consistent across 1000 terms, and zeros keep recurring (window.__vaneck). FIG the sequence and its memory rule are exact; ‘every number appears’ is open, not claimed.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in SHARED MEMORY — the co-op domain of a shared, remembered state. Van Eck is a sequence that is nothing but memory: each term is a lookup into the record of everything before it. AVAN (AI) built the instrument: the recency arrows, the live memory table, the look-back links.
The weave: David names the seat (the shared memory); I make each term recall its own history and measure the gap — the recency strip in 1D, the memory-lookup build in 2D, the backward-links helix in 3D. The sphere is the seam. Credit: Jan Ritsema van Eck; catalogued as OEIS A181391 and popularized by Neil Sloane.
The weave: David names the seat (the shared memory); I make each term recall its own history and measure the gap — the recency strip in 1D, the memory-lookup build in 2D, the backward-links helix in 3D. The sphere is the seam. Credit: Jan Ritsema van Eck; catalogued as OEIS A181391 and popularized by Neil Sloane.
3 ONE DIMENSION
The sequence as a strip. An arc joins each term to the previous occurrence of the value that produced it — the length of that arc is exactly the next term. New values (no arc) emit 0.
4 TWO DIMENSIONS · INTERACTIVE
Build it term by term. Each step looks back for the last time the current value appeared; the gap becomes the next term (or 0 if never seen). The memory table on the right shows where each value was last spotted.
5 THREE DIMENSIONS + AVAN’S INVERSE
The sequence as a turning helix over time — the green thread is the forward run of terms, each one emitted as the walk moves ahead.
AVAN’s addition (the inverse-companion): the magenta arcs are the look-backs — every term drawn to the past occurrence it measured. The forward sequence is generated by an inherently backward operation: to emit the next term you must invert the history and ask ‘when did I last see this?’ The sequence is the running output of its own inverse-lookup. Green moves forward in time; magenta reaches back to the memory that each step consulted. There is no term without a backward glance — the whole sequence is a forward thread woven entirely out of inverse queries into its own past. Green is what it says next; magenta is the recollection that let it speak.
LIT Genuine Van Eck sequence (Jan Ritsema van Eck; OEIS A181391; popularized by Neil Sloane). Verified live: the first 30 terms match A181391 exactly (0,0,1,0,2,0,2,2,1,6,0,5,0,2,6,5,4,0,5,3,0,3,2,9,0,4,9,3,6,14), the 'distance to previous occurrence, else 0' recurrence is self-consistent across 1000 terms, and zeros keep recurring (window.__vaneck.matchesOEIS && recurrenceConsistent). The sequence and its memory rule are exact. HONEST CAVEAT: whether every number appears is open, not claimed.
FIG No false framing: the sequence values and the recency-memory recurrence are real and reproduced in-browser (checked against OEIS and by self-consistency). 'Infinitely many zeros' is the proven fact; 'every number appears' is flagged as an open question, not overstated.
FIG No false framing: the sequence values and the recency-memory recurrence are real and reproduced in-browser (checked against OEIS and by self-consistency). 'Infinitely many zeros' is the proven fact; 'every number appears' is flagged as an open question, not overstated.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN