{"id":"https://openalex.org/W2131975287","doi":"https://doi.org/10.1137/100795425","title":"Nested Recurrence Relations with Conolly-like Solutions","display_name":"Nested Recurrence Relations with Conolly-like Solutions","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2131975287","doi":"https://doi.org/10.1137/100795425","mag":"2131975287"},"language":"en","primary_location":{"id":"doi:10.1137/100795425","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100795425","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1509.02613","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5011644980","display_name":"Alejandro Erickson","orcid":"https://orcid.org/0000-0002-8413-537X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alejandro Erickson","raw_affiliation_strings":["ate@uvic.ca and ruskey@cs.uvic.ca#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ate@uvic.ca and ruskey@cs.uvic.ca#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030862985","display_name":"Abraham Isgur","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Abraham Isgur","raw_affiliation_strings":["abraham.isgur@utoronto.ca and tanny@math.utoronto.ca#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"abraham.isgur@utoronto.ca and tanny@math.utoronto.ca#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109538895","display_name":"Bradley W. Jackson","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bradley W. Jackson","raw_affiliation_strings":["jackson@math.sjsu.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"jackson@math.sjsu.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048017151","display_name":"Frank Ruskey","orcid":"https://orcid.org/0000-0001-5523-7185"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Frank Ruskey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5086012769","display_name":"Stephen M. Tanny","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Stephen M. Tanny","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.0838,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.87639559,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"26","issue":"1","first_page":"206","last_page":"238"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9929999709129333,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9889000058174133,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6816082000732422},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5865253806114197},{"id":"https://openalex.org/keywords/fibonacci-number","display_name":"Fibonacci number","score":0.5635400414466858},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5439630746841431},{"id":"https://openalex.org/keywords/beta","display_name":"BETA (programming language)","score":0.5097207427024841},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.43737897276878357},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3660132884979248}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6816082000732422},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5865253806114197},{"id":"https://openalex.org/C173734053","wikidata":"https://www.wikidata.org/wiki/Q47577","display_name":"Fibonacci number","level":2,"score":0.5635400414466858},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5439630746841431},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.5097207427024841},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.43737897276878357},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3660132884979248},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/100795425","is_oa":false,"landing_page_url":"https://doi.org/10.1137/100795425","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1509.02613","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.02613","pdf_url":"https://arxiv.org/pdf/1509.02613","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.297.4035","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.4035","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://webhome.cs.uvic.ca/~ruskey/Publications/MetaFib/ConollyLikeMay14.pdf","raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1509.02613","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.02613","pdf_url":"https://arxiv.org/pdf/1509.02613","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W299771283","https://openalex.org/W1489341787","https://openalex.org/W1554222672","https://openalex.org/W1848286548","https://openalex.org/W1884798717","https://openalex.org/W1991912396","https://openalex.org/W2040969474","https://openalex.org/W2113034249","https://openalex.org/W2159207128","https://openalex.org/W2752061190","https://openalex.org/W4403215808"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2795454383","https://openalex.org/W2551890212","https://openalex.org/W3100066189","https://openalex.org/W2386746265","https://openalex.org/W3011992704","https://openalex.org/W199458103","https://openalex.org/W2375799027","https://openalex.org/W2369612463","https://openalex.org/W4220747489"],"abstract_inverted_index":{"A":[0,43],"nondecreasing":[1],"sequence":[2,27,147],"of":[3,20,41,72,159,173],"positive":[4,15],"integers":[5,87],"is":[6,28,35,46],"$(\\alpha,\\beta)$-Conolly,":[7],"or":[8],"Conolly-like":[9,60],"for":[10,13,103,163,199],"short,":[11],"if":[12,49,194,197],"every":[14,164],"integer":[16],"m":[17,23],"the":[18,26,38,73,111,134,138,145,170,185],"number":[19],"times":[21],"that":[22,59,94],"occurs":[24],"in":[25,131,184],"$\\alpha":[29,114],"+":[30,141],"\\beta":[31],"r_m$,":[32],"where":[33,113,187],"$r_m$":[34],"1":[36],"plus":[37],"2-adic":[39],"valuation":[40],"m.":[42],"recurrence":[44,70,139,203],"relation":[45],"$(\\alpha,":[47,53,101,108],"\\beta)$-Conolly":[48,54],"it":[50],"has":[51,144],"an":[52,157,160,193],"solution":[55],"sequence.":[56],"We":[57],"discover":[58],"sequences":[61],"often":[62],"appear":[63],"as":[64,148],"solutions":[65,179],"to":[66,128,133,205],"nested":[67,174,202],"(or":[68],"meta-Fibonacci)":[69],"relations":[71],"form":[74],"$A(n)":[75],"=":[76],"\\sum_{i=1}^k":[77],"A(n-s_i-\\sum_{j=1}^{p_i}":[78],"A(n-a_{ij}))$":[79],"with":[80,176],"appropriate":[81],"initial":[82],"conditions.":[83],"For":[84,110],"any":[85],"fixed":[86],"k":[88],"and":[89,116,153,189,195],"$p_1,p_2,\\ldots,":[90],"p_k$":[91],"we":[92,119,155,191],"prove":[93],"there":[95],"are":[96,180],"only":[97,196],"finitely":[98],"many":[99],"pairs":[100],"\\beta)$":[102],"which":[104],"$A(n)$":[105,204],"can":[106],"be":[107,206],"\\beta)$-Conolly.":[109],"case":[112,186],"=0$":[115],"$\\beta":[117],"=1$,":[118],"provide":[120,192],"a":[121,149,200,210],"bijective":[122],"proof":[123],"using":[124],"labeled":[125],"infinite":[126],"trees":[127],"show":[129],"that,":[130],"addition":[132],"original":[135],"Conolly":[136,146],"recurrence,":[137],"$H(n)=H(n-H(n-2))":[140],"H(n-3-H(n-5))$":[142],"also":[143],"solution.":[150],"When":[151],"$k=2$":[152,188],"$p_1=p_2$,":[154,190],"construct":[156],"example":[158],"$(\\alpha,\\beta)$-Conolly":[161],"recursion":[162],"possible":[165],"($\\alpha,\\beta)$":[166],"pair,":[167],"thereby":[168],"providing":[169],"first":[171],"examples":[172],"recursions":[175],"$p_i>1$":[177],"whose":[178],"completely":[181],"understood.":[182],"Finally,":[183],"condition":[198],"given":[201],"$(\\alpha,0)$-Conolly":[207],"by":[208],"proving":[209],"very":[211],"general":[212],"ceiling":[213],"function":[214],"identity.":[215]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}