{"paper":{"title":"Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carsten Peterson, Chi Huynh, Katherine Cordwell, Max Hlavacek, Steven J. Miller, Yen Nhi Truong Vu","submitted_at":"2016-08-31T08:23:38Z","abstract_excerpt":"Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \\dots + c_t H_{n-t}$ where $c_i \\in \\mathbb{N}_0$ for all $i$ and $c_1, c_t \\geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \\in \\mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $\\sigma = (c_1, \\dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $\\sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08764","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}