{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:2U2PDEGQ22FJHL5F7FX4GINTYM","short_pith_number":"pith:2U2PDEGQ","schema_version":"1.0","canonical_sha256":"d534f190d0d68a93afa5f96fc321b3c332aaeef3016640a6c912d1319a00eee8","source":{"kind":"arxiv","id":"1812.11168","version":1},"attestation_state":"computed","paper":{"title":"Combinatorial Identities Deriving From The $N$-th Power Of A $2\\Times 2$ Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin","submitted_at":"2018-12-28T18:52:54Z","abstract_excerpt":"In this paper we give a new formula for the $n$-th power of a $2\\times2$ matrix.\n  More precisely, we prove the following: Let $A= \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right )$ be an arbitrary $2\\times2$ matrix, $T=a+d$ its trace,\n  $D= ad-bc$ its determinant and define \\[ y_{n} :\\,= \\sum_{i=0}^{\\lfloor n/2 \\rfloor}\\binom{n-i}{i}T^{n-2 i}(-D)^{i}. \\] Then, for $n \\geq 1$, \\begin{equation*} A^{n}=\\left ( \\begin{matrix} y_{n}-d \\,y_{n-1} & b \\,y_{n-1} \\\\ c\\, y_{n-1}& y_{n}-a\\, y_{n-1} \\end{matrix} \\right ). \\end{equation*}\n  We use this formula together with an existing formula fo"},"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":"1812.11168","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-28T18:52:54Z","cross_cats_sorted":[],"title_canon_sha256":"519d261b1d2f6abbad419db01d4339f87258effa66b81d511c3bf95447355d96","abstract_canon_sha256":"b1e736e27c1c053d3f6c2fae30230ddecf74a6c345b0de2053cb660669099047"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:16.206850Z","signature_b64":"QRrinX9eIZwV2rRdB7qjVV3Vh8CpqZnlTMqoqE49d2eub/d8j+Ox5+bnmIf1mtoEgU1Fz3JOp9rlifhxCnI4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d534f190d0d68a93afa5f96fc321b3c332aaeef3016640a6c912d1319a00eee8","last_reissued_at":"2026-05-17T23:57:16.206388Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:16.206388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorial Identities Deriving From The $N$-th Power Of A $2\\Times 2$ Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin","submitted_at":"2018-12-28T18:52:54Z","abstract_excerpt":"In this paper we give a new formula for the $n$-th power of a $2\\times2$ matrix.\n  More precisely, we prove the following: Let $A= \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right )$ be an arbitrary $2\\times2$ matrix, $T=a+d$ its trace,\n  $D= ad-bc$ its determinant and define \\[ y_{n} :\\,= \\sum_{i=0}^{\\lfloor n/2 \\rfloor}\\binom{n-i}{i}T^{n-2 i}(-D)^{i}. \\] Then, for $n \\geq 1$, \\begin{equation*} A^{n}=\\left ( \\begin{matrix} y_{n}-d \\,y_{n-1} & b \\,y_{n-1} \\\\ c\\, y_{n-1}& y_{n}-a\\, y_{n-1} \\end{matrix} \\right ). \\end{equation*}\n  We use this formula together with an existing formula 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