Pentagonal sayılar için bazı özdeşlikler ve ağırlıklı toplam formülleri

Year 2025, Issue: 014, 7 - 15, 30.12.2025

Adnan Karataş

Abstract

Bu çalışmada, Euler tarafından incelenmiş ve birçok ünlü matematikçi tarafından ele alınmış bir konu olan beşgensel sayıları araştırdık. Beşgen şekillerden türetilen bu sayı dizisi, aynı zamanda üçgensel sayılarla ilişkisi bakımından da geniş çapta analiz edilmiştir. Genel bilgileri, yineleme bağıntısını ve beşgensel sayılar için Binet formülünü verdikten sonra, elde ettiğimiz sonuçları sunuyoruz. Dizi için çeşitli özdeşlikler sağlıyor ve beşgensel sayıların ağırlıklı, ters ağırlıklı, genelleştirilmiş ağırlıklı ve ters genelleştirilmiş ağırlıklı toplam formüllerini hesaplıyoruz.

Keywords

Beşgensel sayılar , üçüncü dereceden rekürans bağıntıları , genelleştirilmiş ağırlıklı toplama formülleri

Some identities and weighted summation formulas for pentagonal numbers

Abstract

In this study, we explored pentagonal numbers, a topic examined by Euler and discussed by numerous renowned mathematicians. This sequence of numbers, derived from pentagonal shapes, has also been widely analyzed in relation to triangular numbers. After stating the general information, the recurrence relation, and the Binet formula for pentagonal numbers, we present our results. We derive several identities for the sequence, along with weighted, reversed weighted, generalized weighted, and reversed generalized weighted summation formulas.

Keywords

Pentagonal numbers , third order linear recurrence relations , generalized weighted summation formulas

References

• [1] Oeis foundation inc., The On-Line Encyclopedia of Integer Sequences, oeis.org. (accessed Jun 2025).
• [2] G. Andrew, “Euler’s pentagonal number theorem,” Mathematics Magazine, vol 56, pp. 279–284, 1983.
• [3] J. Bell, “A summary of Euler’s work on the pentagonal number theorem,” Archive for history of exact sciences, vol 64, pp. 301–373, 2010.
• [4] G. Çağlayan, “An Identity Relating Triangular and Pentagonal Numbers,” Mathematics Magazine, vol 95, no 4, pp. 405–405, 2022.
• [5] Ü. Sarp, “Visualising connections between types of polygonal number,”, The Mathematical Gazette, vol 107, pp. 56–64, 2023.
• [6] F. Wu and D. Chen, “The Decomposition of Pentagonal Numbers,” Transactions on Computational and Applied Mathematics, vol 4, pp. 113–118, 2024.
• [7] B. Yılmaz, S. Yüksel and N. G. Bilgin, “A Study on Matrix Sequence with Generalized Guglielmo Numbers Components”, Asian Research Journal of Mathematics, vol 20, no 11, pp. 1–25, 2024.
• [8] L. W. Kolitsch and B. Michael, "Interpreting the truncated pentagonal number theorem using partition pairs." The Electronic Journal of Combinatorics, vol 22, pp. 2-55, 2015.
• [9] L. Debnath, "A brief history of partitions of numbers, partition functions and their modern applications." International Journal of Mathematical Education in Science and Technology, vol 47, pp. 329-355, 2016.
• [10] X. Sun, “On Sums of Three Pentagonal Numbers,” The American Mathematical Monthly, vol 123, no 2, pp. 189–191, 2016.
• [11] J. Chahal, M. Griffin and N. Priddis, “When are Multiples of Polygonal Numbers again Polygonal Numbers?,” Hardy-Ramanujan Journal, vol 41, pp. 58–67, 2018.
• [12] S. Yüksel, “Generalized guglielmo numbers: An investigation of properties of triangular, oblong and pentagonal numbers via their third order linear recurrence relations,” Earthline Journal of Mathematical Sciences, vol 9, pp. 1–39, 2022.