{"paper":{"title":"On the number of representations of n as a linear combination of four triangular numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Wang, Zhi-Hong Sun","submitted_at":"2015-07-13T14:59:11Z","abstract_excerpt":"Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For\n  $a,b,c,d,n\\in\\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2\n  +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). In this paper we obtain explicit formulas for $t(a,b,c,d;n)$ in the cases\n  $(a,b,c,d)=(1,2,2,4),\\ (1,2,4,4),\\ (1,1,4,4),\\ (1,4,4,4)$, $(1,3,9,9),\\ (1,1,3,9)$, $(1,3,3,9)$, $(1,1,9,9),\\ (1,9,9,9)$ and $(1,1,1,9).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03485","kind":"arxiv","version":3},"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"}