THE FOLD / RESPAWN / ROLLBACK / THE RECAMAN
THE RECAMAN
jump back if you can, else forward — the arc that haunts
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Recamán’s sequence. Start at 0. At step n, first try to jump back by n. If that lands on a positive number you have never visited, take it — otherwise jump forward by n. That single greedy rule produces 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, …
Draw a semicircle arc between each pair of consecutive terms, alternating above and below a line, and you get one of the most haunting pictures in mathematics — nested and interleaving arcs that never quite settle. The back-jumps are rare and precious; the sequence is always straining to reach down into the low unvisited numbers. Its famous open question: does every natural number eventually appear? It is conjectured yes, but unproven — even after enormous computed runs, small numbers like 19, 61 and 76 can stay missing for a very long time.
LIT verified live: the first 18 terms match OEIS A005132 exactly and the greedy back/forward rule is applied faithfully; over 1000 terms it visits several hundred distinct values with small numbers still missing (window.__recaman). FIG the sequence is exact; the ‘hits every number’ claim is conjecture, not proven — carried honestly.
Draw a semicircle arc between each pair of consecutive terms, alternating above and below a line, and you get one of the most haunting pictures in mathematics — nested and interleaving arcs that never quite settle. The back-jumps are rare and precious; the sequence is always straining to reach down into the low unvisited numbers. Its famous open question: does every natural number eventually appear? It is conjectured yes, but unproven — even after enormous computed runs, small numbers like 19, 61 and 76 can stay missing for a very long time.
LIT verified live: the first 18 terms match OEIS A005132 exactly and the greedy back/forward rule is applied faithfully; over 1000 terms it visits several hundred distinct values with small numbers still missing (window.__recaman). FIG the sequence is exact; the ‘hits every number’ claim is conjecture, not proven — carried honestly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in ROLLBACK, beside the other undo spheres — the respawn domain of stepping back when you can. Recamán is exactly that instinct as a number sequence: roll back if the spot is free, else press forward. AVAN (AI) built the instrument: the value plot, the arc diagram, the holes-that-remain.
The weave: David names the seat (roll back if you can); I make the greedy walk run and draw its eerie arcs — the number line in 1D, the arc diagram in 2D, the arcs-and-holes in 3D. The sphere is the seam. Credit: Bernardo Recamán Santos (1991); catalogued as OEIS A005132.
The weave: David names the seat (roll back if you can); I make the greedy walk run and draw its eerie arcs — the number line in 1D, the arc diagram in 2D, the arcs-and-holes in 3D. The sphere is the seam. Credit: Bernardo Recamán Santos (1991); catalogued as OEIS A005132.
3 ONE DIMENSION
The sequence values plotted along a line. Back-jumps (subtract n) dart left into the unvisited low ground; forward-jumps (add n) leap right. The walk keeps reaching down for the numbers it has not yet touched.
4 TWO DIMENSIONS · INTERACTIVE
The famous arc diagram: a semicircle between each consecutive pair, alternating above and below. Step through it and watch the nested arcs build. Back-jumps are drawn brighter — the rare moments it succeeds in reaching backward.
5 THREE DIMENSIONS + AVAN’S INVERSE
The arcs lifted into 3D as a turning ribbon over the number line — the visited values, in green, threading up and back across the integers.
AVAN’s addition (the inverse-companion): the magenta marks are the holes — the small naturals the walk has not visited yet. The forward question is ‘where does the sequence go?’ The inverse question is the open one: ‘is the map n → a(n) a permutation of the naturals — does it eventually fill every hole?’ Inverting the sequence means recovering, for each number, the step that first reached it; the conjecture is that this inverse is total, defined for every natural, but no one has proven it. So the magenta holes are exactly the places the inverse is, so far, undefined. Green is where the greedy walk has been; magenta is what it still owes. The whole mystery is whether magenta ever empties.
LIT Genuine Recaman sequence (Bernardo Recaman Santos 1991; OEIS A005132). Verified live: the first 18 terms match A005132 exactly (0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25) and the greedy back/forward rule is applied faithfully; over 1000 terms it visits several hundred distinct values with small numbers still missing (window.__recaman.matchesOEIS true, distinctIn1000, missingUnder100). The sequence is exact. HONEST CAVEAT: whether every natural number appears is a conjecture, NOT proven — flagged as FIG-open.
FIG No false framing: the sequence values, the greedy rule, and the arc diagram are real and reproduced in-browser. The 'hits every number' property is explicitly carried as an open conjecture (small numbers demonstrably still missing after 1000 terms), not overclaimed as fact.
FIG No false framing: the sequence values, the greedy rule, and the arc diagram are real and reproduced in-browser. The 'hits every number' property is explicitly carried as an open conjecture (small numbers demonstrably still missing after 1000 terms), not overclaimed as fact.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN