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URN to cite this document: urn:nbn:de:bvb:703-epub-8640-7
Title data
Baumann, Michael Heinrich:
Generalized Triangular Numbers and Combinatorial Explanations.
In: Recreational Mathematics Magazine.
Vol. 12
(June 2025)
Issue 20
.
- pp. 103-119.
ISSN 2182-1976
DOI der Verlagsversion: https://doi.org/10.2478/rmm-2025-0006
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Abstract
The formula for the sums of the first integers, which are known as triangular numbers, is well known and there are many proofs for it: by induction, graphical, by combinatorics, etc. The sum of the first triangular numbers is known as tetrahedral numbers. In this article, we discuss a generalization of triangular and tetrahedral numbers where the number of summation symbols is variable. We repeat results from the literature that state that these so-called generalized triangular numbers can be represented via multicombinations, i.e. combinations with repetitions, and give an illustrative explanation for this formula, which is based on combinatorics. Via high-dimensional illustrations, we show that these generalized triangular numbers are figurate numbers, namely hyper-tetrahedral numbers, see Figure 1. Additionally, we demonstrate that there is a relation between the height and the dimension of these hypertetrahedra, i.e. a series of generalized triangular numbers with fixed dimension and varying height can be represented as such a series with fixed height and varying dimension, and vice versa.
Further data
| Item Type: | Article in a journal |
|---|---|
| Keywords: | Triangular Numbers; Combinatorics; Multicombinations; Figurate Numbers; Hypertetrahedron |
| DDC Subjects: | 500 Science > 510 Mathematics |
| Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Applied Mathematics Research Institutions > Central research institutes > Bayreuth Research Center for Modeling and Simulation - MODUS Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Research Institutions Research Institutions > Central research institutes |
| Language: | English |
| Originates at UBT: | Yes |
| URN: | urn:nbn:de:bvb:703-epub-8640-7 |
| Date Deposited: | 30 Oct 2025 15:54 |
| Last Modified: | 30 Oct 2025 15:54 |
| URI: | https://epub.uni-bayreuth.de/id/eprint/8640 |
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