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When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\\ge0}$ is just the Lucas sequence $(u_n(A,B))_{n\\ge0}$. In this paper, we evaluate the determinants $$\\det[w_{|j-k|}]_{1\\le j,k\\le n}\\ \\ \\text{and}\\ \\ \\det[w_{|j-k+1|}]_{1\\le j,k\\le n}.$$ In particular, we have $$\\det[u_{|j-k|}(A,B)]_{1\\le j,k\\le n}=(-1)^{n-1}u_{n-1}(2A,(B+1)^2).$$ When $B=-1$ and $2\\mid n$, we also determine the characteristic polynomial of the matrix $[w_{j+k}]_{0\\le j,k\\le n-1}$.","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-02-16T14:20:34Z","title":"On determinants involving second-order recurrent sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.08315","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:88aec53c2686f1c5ee4da2a093e0c79b0981268800665521325b09e79fd95772","target":"record","created_at":"2026-07-05T05:43:20Z","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":"2d0f55eff1633505681930ea631adfd900d918127a04a22e25f63140107609d6","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-02-16T14:20:34Z","title_canon_sha256":"51fa789893d46cdd06ec6311bceb50cceff973a8926767c877d0fa6f1df14c49"},"schema_version":"1.0","source":{"id":"2302.08315","kind":"arxiv","version":2}},"canonical_sha256":"639dbbdc1ff33991a3e36be68e4076539167891e39edd3e612fd0cfa042e7206","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"639dbbdc1ff33991a3e36be68e4076539167891e39edd3e612fd0cfa042e7206","first_computed_at":"2026-07-05T05:43:20.745243Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:43:20.745243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Lbz3jih/jJxPe6LKOVkJSXWWB4BI/4wQ4bHnZjO+mD8iY/FcttDDnb8BDJ8hUEhC1y4ATb+dgN3vj1RA9gdpDg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:43:20.745697Z","signed_message":"canonical_sha256_bytes"},"source_id":"2302.08315","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88aec53c2686f1c5ee4da2a093e0c79b0981268800665521325b09e79fd95772","sha256:bae99e4fb57fe4f03261e4525fe10911ab504cc527d0ff0d64d7d7a72b8ca888"],"state_sha256":"95c9107018bf194e339001068d8a0687014f7f8345b15b44844a0c6fa2c416f1"}