{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R703",
  "content_hash": null,
  "slug": "sat-claim-oeis-a030194",
  "type": "claim",
  "title": "The losing positions are OEIS A030194",
  "summary": "OEIS A030194 records the cold positions for the impartial game in which a move subtracts a positive triangular number.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The sequence begins 0, 2, 4, 9, 11, 13, 18, 20, 22, 27, 29, 31, and 51 when the terminal heap 0 is included. This identifies the computed sequence with an existing reference and supplies earlier terminology and links. The OEIS entry is a secondary source; the density-zero proof rests on the polynomial difference theorem.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the losing-position sequence for subtracting positive triangular numbers"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A030194",
      "locator": "OEIS A030194, definition and initial terms"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A030194",
    "locator": "OEIS A030194, definition and initial terms"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R702",
      "title": "Subtract-a-triangular losing positions have density zero",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "subtract-a-triangular-density",
      "title": "subtract a triangular density",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}