Generalized Triangular Numbers and Combinatorial Explanations

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 article1, we discuss a generalizatio...

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Veröffentlicht in:Recreational Mathematics Magazine Jg. 12; H. 20; S. 103 - 119
1. Verfasser: Michael Heinrich Baumann
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Lisbon De Gruyter Brill Sp. z o.o., Paradigm Publishing Services 01.06.2025
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Combinatorics
ISSN:2182-1968, 2182-1976
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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 article1, 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.
AbstractList 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 article1, 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.
Author Michael Heinrich Baumann
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