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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"},"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":"2206.12317","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-06-24T14:32:23Z","cross_cats_sorted":[],"title_canon_sha256":"aca77255224c18e24d616a302cba328c313b2b57f2c765dc7de9018f80804c19","abstract_canon_sha256":"2b67c54cba63227c9dd386f26f9a7c54ff18f1e9251bcf53201fe0657a0643f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:41:21.852119Z","signature_b64":"OmfXNqw8eB8468BHvvJSw5zVXS5SQwsfpXKvVR003D5bY+WoB+uySAw2GgQzlFYwaf2MWF2cAjUDMIzJyVRoAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad5776848787b8ec002d4450cea96c70d1053af95e946315766a9b42c69dbf9c","last_reissued_at":"2026-07-05T05:41:21.851623Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:41:21.851623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evaluations of some Toeplitz-type determinants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Han Wang, Zhi-Wei Sun","submitted_at":"2022-06-24T14:32:23Z","abstract_excerpt":"In this paper we evaluate some Toeplitz-type determinants. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12317","kind":"arxiv","version":8},"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/2206.12317/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":"2206.12317","created_at":"2026-07-05T05:41:21.851691+00:00"},{"alias_kind":"arxiv_version","alias_value":"2206.12317v8","created_at":"2026-07-05T05:41:21.851691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.12317","created_at":"2026-07-05T05:41:21.851691+00:00"},{"alias_kind":"pith_short_12","alias_value":"VVLXNBEHQ64O","created_at":"2026-07-05T05:41:21.851691+00:00"},{"alias_kind":"pith_short_16","alias_value":"VVLXNBEHQ64OYABN","created_at":"2026-07-05T05:41:21.851691+00:00"},{"alias_kind":"pith_short_8","alias_value":"VVLXNBEH","created_at":"2026-07-05T05:41:21.851691+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.09913","citing_title":"Proofs of several OEIS conjectures on determinants and permanents","ref_index":17,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD","json":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD.json","graph_json":"https://pith.science/api/pith-number/VVLXNBEHQ64OYABNIRIM5KLMOD/graph.json","events_json":"https://pith.science/api/pith-number/VVLXNBEHQ64OYABNIRIM5KLMOD/events.json","paper":"https://pith.science/paper/VVLXNBEH"},"agent_actions":{"view_html":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD","download_json":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD.json","view_paper":"https://pith.science/paper/VVLXNBEH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2206.12317&json=true","fetch_graph":"https://pith.science/api/pith-number/VVLXNBEHQ64OYABNIRIM5KLMOD/graph.json","fetch_events":"https://pith.science/api/pith-number/VVLXNBEHQ64OYABNIRIM5KLMOD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD/action/storage_attestation","attest_author":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD/action/author_attestation","sign_citation":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD/action/citation_signature","submit_replication":"https://pith.science/pith/VVLXNBEHQ64OYABNIRIM5KLMOD/action/replication_record"}},"created_at":"2026-07-05T05:41:21.851691+00:00","updated_at":"2026-07-05T05:41:21.851691+00:00"}