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In this paper we consider two classes of sequences: first class is given by the formulae $a_0 = h_1(0), a_n = f(n)a_{n-1} + h_1(n)h_2(n)^n, n>0$, where $f,h_1,h_2 \\in\\mathbb{Z}[X]$, and the second one is defined by $a_n = \\sum_{j=0}^n \\frac{n!}{j!} h(n)^j, n\\in\\mathbb{N}$, where $h\\in\\mathbb{Z}[X]$. Both classes are a generalization of the sequence of derangements. We study such arithmetic properties of these sequences as: periodic"},"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":"1508.01987","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-09T05:40:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e0fa87b37a5e8ece5c2e60e406d9111204c554c35d849869f415a8cfd2471220","abstract_canon_sha256":"d30682b81b5260506f2532ed174972fcaf89fe2a01448bbbe504186859c2d7d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:37.397331Z","signature_b64":"5R6xsJCUioquCPBDPVdeRJ1G73GbSUj9jS/XB55C1YuqmxRMGHIAvF82FVZAXu0QaCEwb6n+bVbuTwii2CydBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b22f68bdafccb11f48e86528b13469f4e9e0335a5acaac5226b0eb4563acc9fd","last_reissued_at":"2026-05-18T01:35:37.396630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:37.396630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic Properties of the Sequence of Derangements and its Generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Piotr Miska","submitted_at":"2015-08-09T05:40:07Z","abstract_excerpt":"The sequence of derangements is given by the formula $D_0 = 1, D_n = nD_{n-1} + (-1)^n, n>0$. It is a classical object appearing in combinatorics and number theory. In this paper we consider two classes of sequences: first class is given by the formulae $a_0 = h_1(0), a_n = f(n)a_{n-1} + h_1(n)h_2(n)^n, n>0$, where $f,h_1,h_2 \\in\\mathbb{Z}[X]$, and the second one is defined by $a_n = \\sum_{j=0}^n \\frac{n!}{j!} h(n)^j, n\\in\\mathbb{N}$, where $h\\in\\mathbb{Z}[X]$. Both classes are a generalization of the sequence of derangements. 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