{"id":"https://openalex.org/W7224197222","doi":"https://doi.org/10.5281/zenodo.23753725","title":"\u2022 \u2022 A Novel Third-Order Recurrence Relation and Its Convergence to the Golden Ratio","display_name":"\u2022 \u2022 A Novel Third-Order Recurrence Relation and Its Convergence to the Golden Ratio","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224197222","doi":"https://doi.org/10.5281/zenodo.23753725"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23753725","doi":"https://doi.org/10.5281/zenodo.23753725","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23753725","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},"volume":null,"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23753725","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158237241","display_name":"Nandkishor prajapat","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"prajapat, nandkishor","raw_affiliation_strings":[": Independent Researcher"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":": Independent Researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5158237241"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.85558443,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/integer-sequences","display_name":"integer sequences","score":0.7870000004768372},{"id":"https://openalex.org/keywords/golden-ratio","display_name":"golden ratio","score":0.8500000238418579},{"id":"https://openalex.org/keywords/lucas-numbers","display_name":"Lucas numbers","score":0.7260000109672546},{"id":"https://openalex.org/keywords/fibonacci-sequence","display_name":"Fibonacci numbers","score":0.4090000092983246}],"concepts":[{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.9271000027656555},{"id":"https://openalex.org/C18802822","wikidata":"https://www.wikidata.org/wiki/Q41690","display_name":"Golden ratio","level":2,"score":0.9146999716758728},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.7454000115394592},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.7300000190734863},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7106999754905701},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.6707000136375427},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6216999888420105},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6093000173568726},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5475999712944031},{"id":"https://openalex.org/C3634175","wikidata":"https://www.wikidata.org/wiki/Q2503280","display_name":"Lucas number","level":3,"score":0.5315999984741211},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48590001463890076},{"id":"https://openalex.org/C154977054","wikidata":"https://www.wikidata.org/wiki/Q1759646","display_name":"Lucas sequence","level":5,"score":0.3831999897956848},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3619000017642975},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34220001101493835},{"id":"https://openalex.org/C61994974","wikidata":"https://www.wikidata.org/wiki/Q2297602","display_name":"Integer sequence","level":3,"score":0.33899998664855957},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.3082999885082245},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.30709999799728394},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.2870999872684479},{"id":"https://openalex.org/C174757984","wikidata":"https://www.wikidata.org/wiki/Q4351325","display_name":"Pisano period","level":5,"score":0.26579999923706055}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23753725","doi":"https://doi.org/10.5281/zenodo.23753725","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23753725","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.5281/zenodo.23753726","doi":"https://doi.org/10.5281/zenodo.23753726","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23753726","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23753725","doi":"https://doi.org/10.5281/zenodo.23753725","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23753725","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},"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/W6949807434","https://openalex.org/W7131671864","https://openalex.org/W4402831227","https://openalex.org/W3126163919","https://openalex.org/W4405957397","https://openalex.org/W7172133403","https://openalex.org/W2488605007","https://openalex.org/W4402767007","https://openalex.org/W2012558139","https://openalex.org/W7164515228"],"abstract_inverted_index":{"This":[0],"paper":[1],"formalizes":[2],"a":[3,10,17],"newly":[4],"discovered":[5],"integer":[6],"sequence":[7,27,67],"defined":[8],"by":[9,23],"unique":[11],"third-order":[12],"linear":[13],"recurrence":[14,38],"relation":[15],"using":[16],"deterministic":[18],"term-skipping":[19],"mechanism.":[20],"Discovered":[21],"independently":[22],"Nandkishor":[24],"Prajapat,":[25],"this":[26,66],"utilizes":[28],"the":[29,37,57,72,79,84],"initial":[30],"seed":[31],"values":[32],"()":[33],"and":[34,59,75],"operates":[35],"under":[36],"rule":[39],"\\(a_n":[40],"=":[41],"2a_{n-2}":[42],"+":[43],"a_{n-3}\\).":[44],"Despite":[45],"utilizing":[46],"architectural":[47],"frameworks":[48],"that":[49,65],"diverge":[50],"completely":[51],"from":[52],"traditional":[53],"second-order":[54],"systems":[55],"like":[56],"Fibonacci":[58],"Lucas":[60,73],"sequences,":[61],"we":[62],"mathematically":[63],"demonstrate":[64],"possesses":[68],"hidden":[69],"cross-connections":[70],"to":[71,78],"numbers":[74],"structurally":[76],"converges":[77],"cosmic":[80],"constant":[81],"known":[82],"as":[83],"Golden":[85],"Ratio":[86],"(\u03c6":[87],"\u2248":[88],"1.618).":[89]},"counts_by_year":[],"updated_date":"2026-10-31T12:09:11.096373","created_date":"2026-10-30T00:00:00"}