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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Signed sums of terms of a sequence


Authors: Feng-Juan Chen and Yong-Gao Chen
Journal: Proc. Amer. Math. Soc. 141 (2013), 1105-1111
MSC (2010): Primary 11A67, 11B50, 11B83, 11P05
Published electronically: August 9, 2012
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Abstract: We give a sufficient and necessary condition on the sequence $ \{a_n\} $ of integers that for any integer $ l\ge 1$, every integer can be represented in the form $ \varepsilon _l a_l+\varepsilon _{l+1} a_{l+1}+\cdots + \varepsilon _ka_k$, where $ \varepsilon _i\in \{-1, 1\}\ (i=l,l+1,\ldots , k)$. This generalizes the known result on integral-valued polynomial values. Moreover, we show that such sequences exist with any growth rate. This answers two problems posed by Bleicher. We also pose several problems for further research.


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Additional Information

Feng-Juan Chen
Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, People’s Republic of China – and – Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China
Email: cfjsz@126.com

Yong-Gao Chen
Affiliation: School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, People’s Republic of China
Email: ygchen@njnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11397-8
PII: S 0002-9939(2012)11397-8
Keywords: Signed sums, sequences, polynomials
Received by editor(s): March 14, 2011
Received by editor(s) in revised form: August 4, 2011
Published electronically: August 9, 2012
Additional Notes: This work was supported by the National Natural Science Foundation of China, Grant No. 11071121 and the Project of Graduate Education Innovation of Jiangsu Province (CXZZ11-0868).
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.