{"id":"https://openalex.org/W13061273","doi":"https://doi.org/10.5951/at.17.1.0033","title":"Polygonal numbers: a study of patterns","display_name":"Polygonal numbers: a study of patterns","publication_year":1970,"publication_date":"1970-01-01","ids":{"openalex":"https://openalex.org/W13061273","doi":"https://doi.org/10.5951/at.17.1.0033","mag":"13061273"},"language":"en","primary_location":{"id":"doi:10.5951/at.17.1.0033","doi":"https://doi.org/10.5951/at.17.1.0033","is_oa":false,"landing_page_url":"https://doi.org/10.5951/at.17.1.0033","pdf_url":null,"source":{"id":"https://openalex.org/S2764413910","display_name":"The Arithmetic Teacher","issn_l":"0004-136X","issn":["0004-136X","2327-0756"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315778","host_organization_name":"National Council of Teachers of Mathematics","host_organization_lineage":["https://openalex.org/P4310315778"],"host_organization_lineage_names":["National Council of Teachers of Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Arithmetic Teacher","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009641438","display_name":"Margaret A. Hervey","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I119234918","display_name":"University of Northern Iowa","ror":"https://ror.org/02h4qpx12","country_code":"US","type":"education","lineage":["https://openalex.org/I119234918"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Margaret A. Hervey","raw_affiliation_strings":["1 University of Northem Iowa, Cedar Falls, Iowa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1 University of Northem Iowa, Cedar Falls, Iowa","institution_ids":["https://openalex.org/I119234918"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084901002","display_name":"Bonnie H. Litwiller","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I119234918","display_name":"University of Northern Iowa","ror":"https://ror.org/02h4qpx12","country_code":"US","type":"education","lineage":["https://openalex.org/I119234918"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bonnie H. Litwiller","raw_affiliation_strings":["1 University of Northem Iowa, Cedar Falls, Iowa"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1 University of Northem Iowa, Cedar Falls, Iowa","institution_ids":["https://openalex.org/I119234918"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I119234918"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"1","first_page":"33","last_page":"38"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.3711000084877014,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.3711000084877014,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12170","display_name":"History and Theory of Mathematics","score":0.2287999987602234,"subfield":{"id":"https://openalex.org/subfields/1207","display_name":"History and Philosophy of Science"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.19699999690055847,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polygonal-numbers","display_name":"polygonal numbers","score":1.0},{"id":"https://openalex.org/keywords/figurate-numbers","display_name":"figurate numbers","score":1.0},{"id":"https://openalex.org/keywords/number-patterns","display_name":"number patterns","score":0.8529999852180481},{"id":"https://openalex.org/keywords/triangular-numbers","display_name":"triangular numbers","score":0.9980000257492065},{"id":"https://openalex.org/keywords/perfect-squares","display_name":"perfect squares","score":1.0},{"id":"https://openalex.org/keywords/pentagonal-numbers","display_name":"pentagonal numbers","score":0.9980000257492065},{"id":"https://openalex.org/keywords/hexagonal-numbers","display_name":"hexagonal numbers","score":0.9959999918937683},{"id":"https://openalex.org/keywords/geometric-patterns","display_name":"geometric patterns","score":0.4650000035762787}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7761132717132568},{"id":"https://openalex.org/C182767506","wikidata":"https://www.wikidata.org/wiki/Q1321169","display_name":"Greeks","level":2,"score":0.6497485637664795},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5299609303474426},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.4845084547996521},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.47511014342308044},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.44507160782814026},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.33299678564071655},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3009227216243744},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.05262908339500427},{"id":"https://openalex.org/C74916050","wikidata":"https://www.wikidata.org/wiki/Q841090","display_name":"Classics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.5951/at.17.1.0033","doi":"https://doi.org/10.5951/at.17.1.0033","is_oa":false,"landing_page_url":"https://doi.org/10.5951/at.17.1.0033","pdf_url":null,"source":{"id":"https://openalex.org/S2764413910","display_name":"The Arithmetic Teacher","issn_l":"0004-136X","issn":["0004-136X","2327-0756"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315778","host_organization_name":"National Council of Teachers of Mathematics","host_organization_lineage":["https://openalex.org/P4310315778"],"host_organization_lineage_names":["National Council of Teachers of Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Arithmetic Teacher","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W94840542","https://openalex.org/W7042368193","https://openalex.org/W4320479107","https://openalex.org/W118591412","https://openalex.org/W282023597","https://openalex.org/W2051707489","https://openalex.org/W7098514853","https://openalex.org/W4220902748","https://openalex.org/W2135385239","https://openalex.org/W4405359040"],"abstract_inverted_index":{"Number":[0],"patterns":[1],"have":[2],"interested":[3],"mathematicians":[4],"from":[5],"the":[6,9,18,31,67,89],"time":[7],"of":[8,13,33,61,69],"ancient":[10],"Greeks.":[11],"Some":[12],"this":[14],"interest":[15],"centered":[16],"around":[17],"relations":[19],"between":[20],"number":[21,60,74],"expressed":[22],"geometrically":[23],"and":[24,53],"algebraically.":[25],"This":[26],"idea":[27],"is":[28,75],"basic":[29],"to":[30,77],"study":[32],"polygonal":[34],"numbers.":[35,48,93],"Polygonal":[36],"numbers,":[37,43,46,50,52],"which":[38],"are":[39],"sometimes":[40],"called":[41],"figurate":[42],"include":[44],"triangular":[45,80,92],"square":[47],"pentagonal":[49],"hexagonal":[51],"so":[54],"on.":[55],"For":[56],"example,":[57],"if":[58],"a":[59,70,79,85],"objects":[62],"can":[63],"be":[64,78],"arranged":[65],"in":[66],"form":[68],"regular":[71],"triangle,":[72],"that":[73],"considered":[76],"number.":[81],"Figure":[82],"I":[83],"shows":[84],"geometrical":[86],"pattern":[87],"for":[88],"first":[90],"five":[91]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}