{"title":"Lucas Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_4","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_4","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"57 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80493356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lehmer Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_5","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_5","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85242713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preliminaries","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_1","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_1","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80150222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Divisibility Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_3","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_3","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91109535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exercises","authors":"Masum Billal, S. Riasat","doi":"10.1017/9781108552332.010","DOIUrl":"https://doi.org/10.1017/9781108552332.010","url":null,"abstract":"Exercise 1: Reduction between Reasoning Tasks We show that A v B if and only if A u ¬B is not satisfiable. If A v B, then in any model I of the TBox, it holds that the AI ⊆ BI . Hence AI ∩ (∆I BI) = ∅. As (A u ¬B)I = AI ∩ (∆I BI), the interpretation of A u ¬B is empty in any model of the TBox, hence A u ¬B is not satisfiable. Conversely, if A is not a subconcept of B there exists a model I of the TBox in which there exists e ∈ AI such that e 6∈ BI . Hence, e ∈ ¬BI , and by definition of conjunction, e ∈ (A ∧ ¬B)I . We have exhibited a model in which (A ∧ ¬B) has a non empty interpretation, and A ∧ ¬B is thus satisfiable. Thus, in order to decide whether A is a subconcept of B, one can check whether A∧¬B is satisfiable. As satisfiability in EL is trivial (every concept is satisfiable) and subsumption is not, there cannot be a reduction from subsumption to satisfiability.","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88948615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Arithmetic Progressions on Conics.","authors":"Abdoul Aziz Ciss, Dustin Moody","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose <i>x</i>-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics <i>ax</i><sup>2</sup> + <i>cy</i><sup>2</sup> = 1 containing arithmetic progressions as long as 8 terms.</p>","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"20 ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5535277/pdf/nihms875076.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35285253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Evaluations of Some Variant Euler Sums","authors":"Hongwei Chen","doi":"10.1090/clrm/035/17","DOIUrl":"https://doi.org/10.1090/clrm/035/17","url":null,"abstract":"In this note we present some elementary methods for the summation of certain Euler sums with terms involving 1 + 1=3 + 1=5 + ¢ ¢ ¢ + 1=(2k i 1):","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"9 1","pages":"0-0"},"PeriodicalIF":0.5,"publicationDate":"2006-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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