{"schema_version": "pages-discovery-record.v1", "record": {"id": "oeis:A010060", "kind": "sequence", "title": "Thue-Morse sequence", "summary": "OEIS sequence with source-backed research connections.", "url": "https://the-omega-institute.github.io/trureturing-pages/oeis/A010060/", "api_url": "https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f3aeacbaef80dc9b34a8e88da4c9b794453b935ed741410acea8f99ba5c513f.json", "status": "external-sequence", "aliases": ["Thue Morse", "Prouhet-Thue-Morse", "binary digit parity", "A010060", "OEIS"], "identifiers": {"oeis": "A010060"}, "evidence": {"source_url": "https://oeis.org/A010060"}}, "relations": [{"source": "oeis:A010060", "target": "result:thue-morse-reduced-abelian-odd", "kind": "studied_object", "scope": "The proved odd recurrence concerns reduced abelian complexity of the Thue-Morse word. A010060 is the underlying binary sequence, not the derived complexity sequence. Even recurrence and nonautomaticity remain outside this result.", "evidence_url": "https://arxiv.org/html/2509.16034v1#S3"}], "neighbors": [{"id": "result:thue-morse-reduced-abelian-odd", "kind": "result", "title": "Thue-Morse reduced abelian complexity: the odd-length recurrence", "summary": "The number of reduced abelian classes at length 2n+1 equals the count at length n+1, for every nonnegative n.", "url": "https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/", "api_url": "https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f11cca9b218ecb086537fbd1533877e565356dab3392ed88327353c8e658c01f.json", "status": "proved", "scope": "This dossier deliberately anchors only the equality\n`rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1)` for every nonnegative integer\n`n`, proposed in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1 and\nprinted page 14 of the verified journal version. The shared Library note\nrecords the journal citation. The source sentence is:\n\n> Although it appears that rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for\n> nonnegative integers n, the problem of determining a full recursion for\n> rho^{ab,red}_t(n) seems to be challenging.\n\n`red(w)` collapses every maximal constant run; reduced abelian complexity\ncounts all length-`n` factors up to rearrangement of their reduced words\nwith equal reduced length. The full recursion and four further questions\nin the source paragraph, namely the nonzero sign of `rho(4n+2)-rho(4n)`, a\nrecursion for `rho(4n)`, equation (11), and non-k-automaticity of sequence\n(10), are deliberately out of scope.", "aliases": ["D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd", "reducedAbelianComplexity_odd", "Combinatorics on words"], "evidence": {"source_url": "https://doi.org/10.48550/arXiv.2509.16034", "source_commit": "b52262a023c4db9bb30a128ebfd80ea4ef4c1a53", "declaration": "D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd.reducedAbelianComplexity_odd", "statement_id": null, "assessment": "kernel-verified", "release_membership": "kernel-verified", "lean_url": "https://the-omega-institute.github.io/trureturing-pages/assets/proofs/thue-morse-reduced-abelian-odd.lean", "source_sha256": "83c07d9a519f88c99485f090e7f988bf5774f32061de9961748a619a22642b18"}}], "index_sha256": "308de5d2dd13f1afe8f6a121329691e0d27eb6872bfcb4f9ebf574a329736090", "research_paths": []}