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The determinants of J_{n} and j_{n} are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that J_{n} and j_{n} are invertible. We also derive the inverses of J_{n} and j_{n}."},"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":"1201.6058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-01-29T17:12:54Z","cross_cats_sorted":[],"title_canon_sha256":"c7f8859e3d005a6e4459ec4c3d07c68048c9a13bfec7de6a83a1a47782ef58cd","abstract_canon_sha256":"e6167b6020bd505c710056587e65c79d6a15a36be8688a3f7e9263e0f8512bb1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:33.797813Z","signature_b64":"EQk9D6FH2Np/XRDi8B1QZ7TiPN5ngVL8zotqQymLEQAyv/oZv8KrU9H8O2cT69/FAwDtqYWxeAr+1faFG4djCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1d73492b7953f16844894c6a09e422fad46208ef6e4981771d59b28cf15e2b2","last_reissued_at":"2026-05-18T04:03:33.797097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:33.797097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Determinants and Inverses of Circulant Matrices with Jacobsthal and Jacobsthal-Lucas Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Durmu\\c{s} Bozkurt, Tin-Yau Tam","submitted_at":"2012-01-29T17:12:54Z","abstract_excerpt":"Let n\\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\\circ(j_{0},j_{1},...,j_{n-1}) be the n\\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The determinants of J_{n} and j_{n} are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that J_{n} and j_{n} are invertible. 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