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For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and 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 reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\\in\\Bbb N$ and $2\\nmid a$. We show that $$t(a,b,c,d;n)=\\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\\ (a,3a,8k+2,8m+6),\\ (a","authors_text":"Min Wang, Zhi-Hong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-02T12:51:36Z","title":"On the number of representations of n as a linear combination of four triangular numbers II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00478","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b2af2aa59954317e8cb710df4bbdc972ea3932f23a881a3b315d4be413b62d00","target":"record","created_at":"2026-05-18T01:24:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1a2a38c3d9e41b4814cd3bb252cfb54491b05a6fc0109f906c32238dca3a223d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-02T12:51:36Z","title_canon_sha256":"cd036b295d6ee217d6d80ec1b880d9e04a57445687b347c6fd88004003328a97"},"schema_version":"1.0","source":{"id":"1511.00478","kind":"arxiv","version":3}},"canonical_sha256":"81ee7bdb5db02e8dc9c4b869ec9e771c55b52a0ba27139768f887df6e240d35f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81ee7bdb5db02e8dc9c4b869ec9e771c55b52a0ba27139768f887df6e240d35f","first_computed_at":"2026-05-18T01:24:45.977184Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:45.977184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qXBCbZMxEgPIRa6XxV3rfjywSAZbmepLy5lHn6/Lc4a7v2fidkSrGe7fDiUYprQimCSMv5hrRxhH5IsUIrw7Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:45.977696Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.00478","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2af2aa59954317e8cb710df4bbdc972ea3932f23a881a3b315d4be413b62d00","sha256:c36ef9964282ab624940cfff5fa0cdd033fdfbfd62a946b411bebf4bbab2c985"],"state_sha256":"52a4e7e100baa67ede2a8b146bd5f331f939924a74d0aa9d3c6c640c02d4fdea"}