{"paper":{"title":"A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tong Niu","submitted_at":"2026-05-14T00:14:56Z","abstract_excerpt":"For the OEIS sequence A032123, the number of length-$2n$ black-and-white strings with $n$ black beads, considered up to reversal, R. J. Mathar contributed in November 2013 the conjectured order-5 P-recursive recurrence \\[ \\begin{aligned} &n(n-1)\\,a(n) - 2(n-1)(3n-4)\\,a(n-1) + 4(2n^{2}-14n+19)\\,a(n-2) &\\qquad + 8(n^{2}+5n-19)\\,a(n-3) - 16(n-3)(3n-10)\\,a(n-4) &\\qquad + 32(n-4)(2n-9)\\,a(n-5) \\;=\\; 0, \\qquad n \\ge 6. \\end{aligned} \\] We give a short proof. Burnside's lemma applied to the reversal action gives the closed form $a(n) = \\tfrac{1}{2}\\bigl(\\binom{2n}{n} + [n \\text{ even}]\\binom{n}{n/2}\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The closed form a(n) = 1/2 (binomial(2n,n) + [n even] binomial(n,n/2)) correctly counts the orbits under the reversal group action, which rests on the standard application of Burnside's lemma to the two-element group.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9d6bc1ddc51ef01e7399af09c71887d4957c80f9ac395a25771b2956caf29f56"},"source":{"id":"2605.14213","kind":"arxiv","version":1},"verdict":{"id":"bc5cb04f-e5d6-4e61-ab91-a0bc6eacda5d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:51:24.114460Z","strongest_claim":"Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation.","one_line_summary":"The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The closed form a(n) = 1/2 (binomial(2n,n) + [n even] binomial(n,n/2)) correctly counts the orbits under the reversal group action, which rests on the standard application of Burnside's lemma to the two-element group.","pith_extraction_headline":"Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal."},"references":{"count":11,"sample":[{"doi":"","year":null,"title":"Burnside,Theory of Groups of Finite Order, Cambridge University Press, 1897","work_id":"0109df99-6c74-40ae-8b96-1fc5e07fbce9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"S. Chen, M. Kauers, C. Koutschan, X. Li, R.-H. Wang and Y. Wang,Non-minimality of minimal telescopers explained by residues, arXiv:2502.03757, 2025","work_id":"0b8abde8-788b-48a1-93ee-3e02d9581456","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Fried,Proofs of some conjectures from the OEIS","work_id":"5fb607ef-406b-4aff-ae7a-71aa50508bf2","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"S. Fried,Proofs of several conjectures from the OEIS, J. Integer Seq.28(2025), Article 25.4.3. 10 TONG NIU","work_id":"fdc55654-e319-47dc-9ff2-c6deee479c49","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"M. Kauers and C. Koutschan,A list of guessed but unproven holonomic recurrences in the OEIS, arXiv:2303.02793, 2023","work_id":"6d053669-aa63-4eeb-b76d-b87c3501d616","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"5522a88d941f37fff786548c40e88afb8f28301ab1e08c1020d80037bf7d9de4","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}