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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 $[w_{j+k}]_{0\\le j,k\\le n-1}$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2302.08315","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-02-16T14:20:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"51fa789893d46cdd06ec6311bceb50cceff973a8926767c877d0fa6f1df14c49","abstract_canon_sha256":"2d0f55eff1633505681930ea631adfd900d918127a04a22e25f63140107609d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:43:20.745697Z","signature_b64":"Lbz3jih/jJxPe6LKOVkJSXWWB4BI/4wQ4bHnZjO+mD8iY/FcttDDnb8BDJ8hUEhC1y4ATb+dgN3vj1RA9gdpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"639dbbdc1ff33991a3e36be68e4076539167891e39edd3e612fd0cfa042e7206","last_reissued_at":"2026-07-05T05:43:20.745243Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:43:20.745243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On determinants involving second-order recurrent sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2023-02-16T14:20:34Z","abstract_excerpt":"Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\\ldots$. 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 $[w_{j+k}]_{0\\le j,k\\le n-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.08315","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.08315/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2302.08315","created_at":"2026-07-05T05:43:20.745300+00:00"},{"alias_kind":"arxiv_version","alias_value":"2302.08315v2","created_at":"2026-07-05T05:43:20.745300+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.08315","created_at":"2026-07-05T05:43:20.745300+00:00"},{"alias_kind":"pith_short_12","alias_value":"MOO3XXA76M4Z","created_at":"2026-07-05T05:43:20.745300+00:00"},{"alias_kind":"pith_short_16","alias_value":"MOO3XXA76M4ZDI7D","created_at":"2026-07-05T05:43:20.745300+00:00"},{"alias_kind":"pith_short_8","alias_value":"MOO3XXA7","created_at":"2026-07-05T05:43:20.745300+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO","json":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO.json","graph_json":"https://pith.science/api/pith-number/MOO3XXA76M4ZDI7DNPTI4QDWKO/graph.json","events_json":"https://pith.science/api/pith-number/MOO3XXA76M4ZDI7DNPTI4QDWKO/events.json","paper":"https://pith.science/paper/MOO3XXA7"},"agent_actions":{"view_html":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO","download_json":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO.json","view_paper":"https://pith.science/paper/MOO3XXA7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2302.08315&json=true","fetch_graph":"https://pith.science/api/pith-number/MOO3XXA76M4ZDI7DNPTI4QDWKO/graph.json","fetch_events":"https://pith.science/api/pith-number/MOO3XXA76M4ZDI7DNPTI4QDWKO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO/action/storage_attestation","attest_author":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO/action/author_attestation","sign_citation":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO/action/citation_signature","submit_replication":"https://pith.science/pith/MOO3XXA76M4ZDI7DNPTI4QDWKO/action/replication_record"}},"created_at":"2026-07-05T05:43:20.745300+00:00","updated_at":"2026-07-05T05:43:20.745300+00:00"}