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Let $n>1$ be an integer. We prove the following two basic identities: \\begin{align*} \\det{[j-k+\\delta_{jk}]_{1\\leq j,k\\leq n}}&=1+\\frac{n^2(n^2-1)}{12}, \\\\ \\det{[|j-k|+\\delta_{jk}]_{1\\leq j,k\\leq n}}&= \\begin{cases} \\frac{1+(-1)^{(n-1)/2}n}{2}&\\text{if}\\ 2\\nmid n,\\\\ \\frac{1+(-1)^{n/2}}{2}&\\text{if}\\ 2\\mid n, \\end{cases} \\end{align*} where $\\delta_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\\not=0$ and $a^2\\not=4b$, and the sequence $(w_m)_{m\\in\\mathbb Z}$ with $w_{k+1}=aw_k-bw_{k-1}$ for all $k\\in\\mathbb Z$, we est","authors_text":"Han Wang, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-06-24T14:32:23Z","title":"Evaluations of some Toeplitz-type determinants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12317","kind":"arxiv","version":8},"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":"sha256:5889ceb15f3984020901899d4691b0e52daa30bd75a13e0902123f203eedf653","target":"record","created_at":"2026-07-05T05:41:21Z","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":"2b67c54cba63227c9dd386f26f9a7c54ff18f1e9251bcf53201fe0657a0643f1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-06-24T14:32:23Z","title_canon_sha256":"aca77255224c18e24d616a302cba328c313b2b57f2c765dc7de9018f80804c19"},"schema_version":"1.0","source":{"id":"2206.12317","kind":"arxiv","version":8}},"canonical_sha256":"ad5776848787b8ec002d4450cea96c70d1053af95e946315766a9b42c69dbf9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad5776848787b8ec002d4450cea96c70d1053af95e946315766a9b42c69dbf9c","first_computed_at":"2026-07-05T05:41:21.851623Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:41:21.851623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OmfXNqw8eB8468BHvvJSw5zVXS5SQwsfpXKvVR003D5bY+WoB+uySAw2GgQzlFYwaf2MWF2cAjUDMIzJyVRoAg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:41:21.852119Z","signed_message":"canonical_sha256_bytes"},"source_id":"2206.12317","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5889ceb15f3984020901899d4691b0e52daa30bd75a13e0902123f203eedf653","sha256:9fb8c33692e53dd04aee7f4a8be4195bb1b97e823db035f0ef68a1bdbe4639cf"],"state_sha256":"600660d88c3d61ffea8b46ab12c77464db0d1e5f101aadf455337d8964863915"}