ProofAtlas

Integer sequences and coverage

Recamán’s Sequence Coverage Conjecture

Collaboration beta

Does the standard Recamán sequence visit every nonnegative integer? Addition may repeat a value; the question concerns coverage. The source leaves both coverage and permanent omission unresolved.

a0=0,an=an-1-n,&an-1-n≥0andan-1-n∉Sn-1,an-1+n,&otherwise,Sn={a0,…,an}.
Known results and sources
Illustration: At step 24, subtraction from 18 would be negative, so addition returns to 42, a value already visited at step 20.
Illustration: The actual step a₂₄ = 18 + 24 = 42 repeats a₂₀. Repetition is allowed on the addition branch.

Research problem

Exact mathematical statement

Full objective: prove that every nonnegative integer occurs in the standard Recamán sequence, or disprove this by certifying a permanently omitted integer. The present work has neither a coverage proof nor a permanent-omission proof. The source reports exact reductions and local avoidance theorems, and describes its finite certificates as reproducible. These results and computations have not been independently checked here and do not establish a completed resolution.

The target is the standard sequence

a0=0,an=an-1-n,&an-1-n≥0andan-1-n∉Sn-1,an-1+n,&otherwise,Sn={a0,…,an}.a_0=0,\qquad a_n=\begin{cases} a_{n-1}-n,& a_{n-1}-n\ge0\text{ and }a_{n-1}-n\notin S_{n-1},\\ a_{n-1}+n,&\text{otherwise}, \end{cases} \qquad S_n=\{a_0,…,a_n\}.

This is OEIS A005132; the definition and permission for repetitions on the addition branch are confirmed by the primary references [SRC-01] and [SRC-03]. The target is coverage, not injectivity. Repetition on addition is part of the problem and may not be removed by an optimization or comparison argument.

Problem infographic

Problem at a glance

Illustration: Actual small Recamán choices and a bounded prefix show how legal subtraction differs from addition that may repeat; coverage of every integer stays open.
Illustration: Small recurrence examples and temporarily unvisited integers illustrate the rule. They do not prove that every integer is eventually visited or that any integer is permanently omitted.

Current mathematical picture

Where work on Recamán’s Sequence Coverage Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAssume that adding occupied values can only improve local safety

At (q,d,j)=(3,13,0), the history containing (3,13), (1,7), and (1,10), with other queried points unoccupied, is safe; adding (2,12) yields (3,13)→(4,10)→(3,6)→(2,3)→(1,1), whose next subtraction hits. These are consistent artificial histories, not asserted actual prefixes. The one-interval and two-level theorems rely on specified positive occupied sets and certify the same entrance. Their stated hypotheses remain available even though arbitrary-history monotonicity fails.

Route status · Narrowed route
Main reductionA conditional route to permanent omission

Starting from the stated untouched prefix at N0=30000000, a sufficient positive certificate of either kind at every future active entrance would permanently protect G=[27662800,27662934] and disprove coverage. The source reports the implication, while explicitly leaving its universal regeneration premise unproved.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeEstablish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.Task status · Ready to work on

Work mapped so far

Recamán’s Sequence Coverage Conjecture in numbers

1.2kretained lines of mathematical investigation1,229 in the current working snapshot
Argument development
719 · 59%
Explored or eliminated routes
71 · 6%
Computational analysis
159 · 13%
Open obligations
206 · 17%
Definitions and setup
74 · 6%
9selected mapped statements2routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byN. J. A. SloaneProposed1991Status checkedSep 7, 2026
Source recordedSep 7, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

14 selected steps

Scroll horizontally to explore the route

Working route overview for Recamán’s Sequence Coverage ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Coverage of every nonnegative integer, with repetitions permitted on addition — Depends on missing premiseCoverage of everynonnegative integer, withrepetitions…A conditional route to permanent omission — Depends on missing premiseA conditional route topermanent omissionCurrent reduction — Depends on missing premiseCurrent reductionExact wide-gap proposal criterion — Depends on missing premiseExact wide-gap proposalcriterionMarked-gap coupling and late clearing — Depends on missing premiseMarked-gap coupling and lateclearingBounded local decision, with history retained — Depends on missing premiseBounded local decision, withhistory retainedClosing target — Depends on missing premiseClosing targetFinite certificates through index 10^13 — Depends on missing premiseFinite certificates throughindex 10^13Positive certificates protect one entrance — Depends on missing premisePositive certificatesprotect one entranceAssume that adding occupied values can only improve local safety — stoppedAssume that adding occupiedvalues can only improvelocal…Carry a certificate unchanged to every later entrance merely because the visited set grows — stoppedCarry a certificateunchanged to every laterentrance…Establish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history. — OpenEstablish perpetualregeneration of aone-interval…Construct a globally valid transition for incoming occupancy profiles, including consistency and generated history, instead of treating the finite local query mask as a closed state. — OpenConstruct a globally validtransition for incomingoccupancy…An alternative coverage route must force a hitting profile from actual Recamán history; realizability of a safe or dangerous artificial local state alone is insufficient. — OpenAn alternative coverageroute must force a hittingprofile…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded
Narrowed routeAssume that adding occupied values can only improve local safety

At (q,d,j)=(3,13,0), the history containing (3,13), (1,7), and (1,10), with other queried points unoccupied, is safe; adding (2,12) yields (3,13)→(4,10)→(3,6)→(2,3)→(1,1), whose next subtraction hits. These are consistent artificial histories, not asserted actual prefixes. The one-interval and two-level theorems rely on specified positive occupied sets and certify the same entrance. Their stated hypotheses remain available even though arbitrary-history monotonicity fails.

Route status · Narrowed route
Narrowed routeCarry a certificate unchanged to every later entrance merely because the visited set grows

The source reports that H=[167526,167661] had a same-entrance certificate at N=471296, yet a later entrance at N′=65854466988 led to a hit at 167659 and then the gap bottom 167526. The old occupied interval remained at the wrong location for the later phase. A renewal theorem that produces the required occupied intervals at each new entrance remains a candidate route; the source explicitly leaves this regeneration step open.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Establish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.Suggested move: Suggested next step: identify how newly occupied intervals are born and transported to the next entrance scale, then state an inductive renewal invariant with explicit initialization and preservation clauses.
Ready to work on
02
Construct a globally valid transition for incoming occupancy profiles, including consistency and generated history, instead of treating the finite local query mask as a closed state.Suggested move: Suggested next step: trace the origin of every bit needed at the next entrance and prove that the retained state determines it; reject artificial completions unless their reachability from the actual prefix is established.
Ready to work on
03
An alternative coverage route must force a hitting profile from actual Recamán history; realizability of a safe or dangerous artificial local state alone is insufficient.Suggested move: Suggested next step: formulate an actual-history realization theorem for a dangerous profile and show how it would force coverage, keeping this alternative separate from the positive-certificate disproof route.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen conjecture

It remains unknown whether the standard Recamán sequence reaches every nonnegative integer. Starting at zero, subtraction is used only for a nonnegative value not previously visited; otherwise addition is used and may repeat a value.

[1][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Computational resultChaffin reports computing beyond 10^612 terms while 852655 remains the smallest unvisited nonnegative integer. This finite observation does not show that it is never visited.[2]
3 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusRecamán’s Sequence Coverage Conjecture
Related problemThree greedy cousins of Recamán's sequence

Alekseyev and coauthors study distinct greedy constructions with different operation rules. Results for those sequences do not settle coverage for the original recurrence.

[3]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasOEIS standard Recamán sequence data

    The entry supplies the recurrence and finite terms, including repeated values. Its linked tables were not reproduced.

    [1]
  • computation · not independently reproducedChaffin's 2026 Recamán computation

    The author's page reports a finite computational horizon and missing-value information. The implementation and result files were not executed or checked.

    [2]

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction3 of 93
  • lemma3 of 93
  • equivalence1 of 91
  • computational claim1 of 91
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
  • retained route statementCoverage of every nonnegative integer, with repetitions permitted on addition
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementMarked-gap coupling and late clearingintermediate
  • retained route statementExact wide-gap proposal criterionintermediate
  • retained route statementBounded local decision, with history retainedintermediate
  • retained route statementPositive certificates protect one entranceintermediate
  • retained route statementA conditional route to permanent omissionintermediate
  • retained route statementFinite certificates through index 10^13intermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureAssume that adding occupied values can only improve local safetyreported failure
  • Useful failureCarry a certificate unchanged to every later entrance merely because the visited set growsreported failure
  • Research targetEstablish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.open
  • Research targetConstruct a globally valid transition for incoming occupancy profiles, including consistency and generated history, instead of treating the finite local query mask as a closed state.open
  • Research targetAn alternative coverage route must force a hitting profile from actual Recamán history; realizability of a safe or dangerous artificial local state alone is insufficient.open
  • ComputationThe source reports two finite engines reaching index 10^13, with direct comparison through 10^6, checkpoint comparisons, and exact incoming profiles for 23 active entrance groups between indices 30000000 and 10^13.Among the 23 observed groups, 22 have a one-interval certificate and 10 have a two-level certificate; their union covers 23 groups. The candidate gap remains missing at the finite horizon, without a theorem about later entrances. · reported unreproduced
  • Narrowed routeAssume that adding occupied values can only improve local safetyAt (q,d,j)=(3,13,0), the history containing (3,13), (1,7), and (1,10), with other queried points unoccupied, is safe; adding (2,12) yields (3,13)→(4,10)→(3,6)→(2,3)→(1,1), whose next subtraction hits. These are consistent artificial histories, not asserted actual prefixes. The one-interval and two-level theorems rely on specified positive occupied sets and certify the same entrance. Their stated hypotheses remain available even though arbitrary-history monotonicity fails.
  • Narrowed routeCarry a certificate unchanged to every later entrance merely because the visited set growsThe source reports that H=[167526,167661] had a same-entrance certificate at N=471296, yet a later entrance at N′=65854466988 led to a hit at 167659 and then the gap bottom 167526. The old occupied interval remained at the wrong location for the later phase. A renewal theorem that produces the required occupied intervals at each new entrance remains a candidate route; the source explicitly leaves this regeneration step open.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeEstablish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

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Prepared starting pointEstablish perpetual regeneration of a one-interval or two-level positive certificate along the actual future Recamán history.

Recamán’s Sequence Coverage Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Does the standard Recamán sequence visit every nonnegative integer? Addition may repeat a value; the question concerns coverage. The source leaves both coverage and permanent omission unresolved.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A005132 — Recamán's sequencesoftware or dataset · The OEIS Foundation Inc. · OEIS A005132 · accessed Sep 7, 2026
  2. 2
    The Recamán Sequenceauthoritative webpage · Ben Chaffin · 2026-02-07 · accessed Sep 7, 2026
  3. 3
    Three Cousins of Recaman's Sequencepeer reviewed result · Max A. Alekseyev, Joseph Samuel Myers, Richard Schroeppel, S. R. Shannon, N. J. A. Sloane, Paul Zimmermann · The Fibonacci Quarterly · 2022 · ARXIV 2004.14000v2 · DOI 10.1080/00150517.2022.12427470 · accessed Sep 7, 2026

Important qualifications

  • The workspace title is packet-defined. Proposer and year reflect Sloane's own retrospective OEIS statement about the coverage conjecture; the original 1991 correspondence was not inspected and no priority claim is made.
  • The sequence's inventor is not inferred to be the coverage conjecture's proposer. The OEIS also records Sloane's later doubt about complete coverage.
  • Chaffin's very large finite computation is an attributed author report; no code, linked result table, or computational certificate was downloaded or reproduced.
  • No exact formalization was established by this bounded search. This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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