Generalized Triangular Numbers and Combinatorial Explanations

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Baumann, Michael Heinrich:
Generalized Triangular Numbers and Combinatorial Explanations.
In: Recreational Mathematics Magazine. Bd. 12 (2025) Heft 20 . - S. 103-119.
ISSN 2182-1976
DOI: 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.