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We give three short proofs. The first is two lines: subtract the defining recurrence at adjacent indices and the constant cancels (we call this homogenisation). The second reads off the same relation from the exponential generating function $F(x) = ","authors_text":"Tong Niu","cross_cats":[],"headline":"The sequence a(n) = n a(n-1) + 1 with a(0) = 0 satisfies a(n) - (n+1) a(n-1) + (n-1) a(n-2) = 0 for all n >= 2.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T00:36:49Z","title":"Three short proofs of Mathar's 2014 conjecture for OEIS A002627"},"references":{"count":11,"internal_anchors":0,"resolved_work":11,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"M. Kauers and C. 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Software20(1994), 163–177","work_id":"fa8f75a2-1770-4540-a4b6-59f7d45b40b2","year":1994},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Singh,OnL(m, n)and the Bernoulli and Eulerian numbers, Math","work_id":"b1cf654e-55a6-478e-b56e-db0dd8c3b22c","year":1952}],"snapshot_sha256":"7910ecd1e6f3480b2f9506bd48370ca7eabb4052b1c4c56e0071bdc5d0284698"},"source":{"id":"2605.15500","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T15:47:43.264907Z","id":"43171870-8907-4fc0-bc9f-3e34b8dbd2ad","model_set":{"reader":"grok-4.3"},"one_line_summary":"Three short elementary proofs establish the conjectured homogeneous recurrence for OEIS A002627 from its defining inhomogeneous recurrence.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"The sequence a(n) = n a(n-1) + 1 with a(0) = 0 satisfies a(n) - (n+1) a(n-1) + (n-1) a(n-2) = 0 for all n >= 2.","strongest_claim":"The sequence a(n) defined by a(n) = n a(n-1) + 1, a(0) = 0 satisfies a(n) - (n+1) a(n-1) + (n-1) a(n-2) = 0 for all n >= 2.","weakest_assumption":"The first-order recurrence a(k) = k a(k-1) + 1 holds exactly for every integer k >= 1, so that subtraction at adjacent indices is valid and produces no boundary or remainder terms."}},"verdict_id":"43171870-8907-4fc0-bc9f-3e34b8dbd2ad"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:89148723dfd575f0af054009d08791795923e4864eaf7397905c585dd26dca77","target":"record","created_at":"2026-05-20T00:01:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e2764533384f774269f767e12f34025b5d287aeb7b89a8c5e29250c2e5f24110","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T00:36:49Z","title_canon_sha256":"e0d9c94d9e3dc39e18cd2e415b03d91da827af5d74021cf6d6d7ab95cd1d0950"},"schema_version":"1.0","source":{"id":"2605.15500","kind":"arxiv","version":1}},"canonical_sha256":"1a82742dbd98ebdf0ef8844109d34715fce10c2c8f811c4dc3a0e9ba4aa2d8cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a82742dbd98ebdf0ef8844109d34715fce10c2c8f811c4dc3a0e9ba4aa2d8cb","first_computed_at":"2026-05-20T00:01:01.860347Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:01:01.860347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lSigArlTC5au/KgkfYaf3QiAstZyB63IlrD9+Cr0CKHMY2NnGF8asBBUMPMtDunng96Z507h9U9QSQPHeg4JDg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:01:01.861138Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15500","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89148723dfd575f0af054009d08791795923e4864eaf7397905c585dd26dca77","sha256:3f8f9c78a4e69ad5ab2b0644a826dcbd97f18c1dfe0b0116da53b310d68cf2b0"],"state_sha256":"45dc9d4600d6dc7444ac855305af62faeb864be2274abd537ee389a77ecc868d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0rYBZIhCOnzK9V7aLuY3tZH4smSHeNfvDWPFfZ1CwzJHriLhKxvCtFilRVZENJ7opPb+2VYuxBnxkZt6tWuXAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T02:58:43.584606Z","bundle_sha256":"f5bc4325d9949c188018ef6f02cab05ac2dcfd89c95df740a4ac75ff9a006e65"}}