On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers
A trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introd...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2017/4515249 |
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| author | Somphong Jitman Chakrit Phongthai |
| author_facet | Somphong Jitman Chakrit Phongthai |
| author_sort | Somphong Jitman |
| collection | DOAJ |
| description | A trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introduced. For each positive integer m, a positive integer N is called an m-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m. Properties of m-trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where m=2, the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed 2-trapezoidal number N, the ways and the number of ways to write N as an arithmetic series with common difference 2 have been determined. Some remarks on 3-trapezoidal numbers have been provided as well. |
| format | Article |
| id | doaj-art-f90eb496191b468caffd020c50e055ce |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-f90eb496191b468caffd020c50e055ce2025-02-03T06:00:53ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252017-01-01201710.1155/2017/45152494515249On the Characterization and Enumeration of Some Generalized Trapezoidal NumbersSomphong Jitman0Chakrit Phongthai1Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, ThailandDepartment of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, ThailandA trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introduced. For each positive integer m, a positive integer N is called an m-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m. Properties of m-trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where m=2, the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed 2-trapezoidal number N, the ways and the number of ways to write N as an arithmetic series with common difference 2 have been determined. Some remarks on 3-trapezoidal numbers have been provided as well.http://dx.doi.org/10.1155/2017/4515249 |
| spellingShingle | Somphong Jitman Chakrit Phongthai On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers International Journal of Mathematics and Mathematical Sciences |
| title | On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers |
| title_full | On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers |
| title_fullStr | On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers |
| title_full_unstemmed | On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers |
| title_short | On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers |
| title_sort | on the characterization and enumeration of some generalized trapezoidal numbers |
| url | http://dx.doi.org/10.1155/2017/4515249 |
| work_keys_str_mv | AT somphongjitman onthecharacterizationandenumerationofsomegeneralizedtrapezoidalnumbers AT chakritphongthai onthecharacterizationandenumerationofsomegeneralizedtrapezoidalnumbers |