{"paper":{"title":"Picturesque convolution-like recurrences and partial sums' generation","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrius Grigutis, Ignas Gasparavi\\v{c}ius, Juozas Petkelis","submitted_at":"2025-07-31T14:58:18Z","abstract_excerpt":"Let ${\\pmb b}=\\{b_0,\\,b_1,\\,\\ldots\\}$ be the known sequence of numbers such that $b_0\\neq0$. In this work, we develop methods to find another sequence ${\\pmb a}=\\{a_0,\\,a_1,\\,\\ldots\\}$ that is related to ${\\pmb b}$ as follows: $a_n=a_0\\,b_{n+m}+a_1\\,b_{n+m-1}+\\ldots+a_{n+m}\\,b_0$, $n\\in\\mathbb{N}\\cup\\{0\\}$, $m\\in\\mathbb{N}$. We show the connection of $\\lim_{n\\to\\infty}a_n$ with $a_0,\\,a_1,\\,\\ldots,\\,a_{m-1}$ and provide varied examples of finding the sequence ${\\pmb a}$ when ${\\pmb b}$ is given. We demonstrate that the sequences ${\\pmb a}$ may exhibit pretty patterns in the plane or space. Als"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.23619","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.23619/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}