{"title":"关于广义Lucas序列的一些新结果","authors":"D. Andrica, O. Bagdasar, George C. Ţurcaş","doi":"10.2478/auom-2021-0002","DOIUrl":null,"url":null,"abstract":"Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.","PeriodicalId":55522,"journal":{"name":"Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica","volume":"90 1","pages":"17 - 36"},"PeriodicalIF":0.8000,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On some new results for the generalised Lucas sequences\",\"authors\":\"D. Andrica, O. Bagdasar, George C. Ţurcaş\",\"doi\":\"10.2478/auom-2021-0002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.\",\"PeriodicalId\":55522,\"journal\":{\"name\":\"Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica\",\"volume\":\"90 1\",\"pages\":\"17 - 36\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2021-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2478/auom-2021-0002\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2478/auom-2021-0002","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On some new results for the generalised Lucas sequences
Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.
期刊介绍：
This journal is founded by Mirela Stefanescu and Silviu Sburlan in 1993 and is devoted to pure and applied mathematics. Published by Faculty of Mathematics and Computer Science, Ovidius University, Constanta, Romania.
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