Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular

Francisco Hedyleno Coelho Bezerra; Francisco Regis Vieira Alves; Eudes Antonio Costa

doi:10.33434/cams.2016687

Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular

Abstract

A study of the multi-figurate numbers that are simultaneously triangular and centered triangular is presented. Starting from the generalized Pell equation associated with this class of multi-figurate numbers, we describe the corresponding sequence and investigate its algebraic properties. In particular, we obtain a companion matrix representation for the sequence. In addition, we derive several identities of Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne types satisfied by this sequence. These results provide a systematic description of the algebraic structure associated with the sequence of numbers that are simultaneously triangular and centered triangular.

Keywords

Triangular numbers, Centered triangular numbers, Pell equation, Recurrence relations

References

R. P. Agarwal, Pythagoreans figurative Numbers: The beginning of number theory and summation of series, J. Appl. Math. Phys., 9(8) (2021), 2038–2113. https://doi.org/10.4236/jamp.2021.98132
E. Deza, M. M. Deza, Figurate Numbers, World Scientific, Singapore, 2012.
R. B. Nelsen, Multi-polygonal numbers, Math. Mag., 89(3) (2016), 159–164. https://doi.org/10.4169/math.mag.89.3.159
S. J. Schlicker, Numbers simultaneously polygonal and centered polygonal, Math. Mag., 84(5) (2011), 339–350. https://doi.org/10.4169/math.mag.84.5.339
A. Emin, Some multi-figurate numbers in terms of generalized Fibonacci and Lucas numbers, Aligarh Bull. Math., 42(1) (2023), 107–123.
F. R. V. Alves, F. H. C. Bezerra, P. M. M. C. Catarino, et al., Números multifigurados e a relação de recorrência, Prof. Mat. Online, 14(3) (2026), 431–457. https://doi.org/10.21711/2319023x2026/pmo1426
F. H. C. Bezerra, F. R. V. Alves, E. A. da Costa, et al., Uma nota sobre os números multifigurados quadrados-nonagonais, Intermaths, 6(1) (2025), 93–106.
E. Gebeş, A. Emin, U. Sarp, On triangular pentagonal and square pentagonal numbers, In M. Çevik (Ed.), Scientific Research Reports, Duvar Publications, 2024, pp. 47–56.
R. Sivaraman, Octagonal–pentagonal numbers and an interesting puzzle, Bull. Math. Stat. Res., 10(1) (2022), 16–20.
R. Sivaraman, Square–pentagonal numbers, Int. J. Multidiscip. Res. Mod. Educ., 8(1) (2022), 1–2. https://doi.org/10.5281/zenodo.5829440

L. Tamilarasi, R. Sivaraman, Centered triangular, triangular, square, centered square, hexagonal and heptagonal numbers, Adv. Nonlinear Var. Inequalities, 27(1) (2024), 275–285. https://doi.org/10.52783/anvi.v27.436
E. D’Amora, G. Vincenzi, Numbers that are simultaneously centered polygonal, J. Korean Math. Soc., 63(3) (2026), 619–637. https://doi.org/10.4134/JKMS.j250282
F. R. V. Alves, Numeros figurados gregos: o caso dos triangulares, quadrados e sua recorrˆencia,Rev. Hist. Mat. Profr. (Online), 10(1) (2024), 1–12.
D. M. Burton, The History of Mathematics: An Introduction, McGraw-Hill, New York, 2011.
H. Eves, Introdução à História da Matemática, 5th ed., Editora da Unicamp, Campinas, 2011.
T. Roque, Historia da Matematica: Uma Visão Critica, Desfazendo Mitos e Lendas, Jorge Zahar (Ed.), Rio de Janeiro, 2012.
U. Sarp, Visualising connections between types of polygonal number, Math. Gaz., 107(568) (2023), 56–64. https://doi.org/10.1017/mag.2023.7
F. E. Barros, Engenharia didática sobre números figurados k-dimensionais: Relações recorrentes, termo geral, função geradora e representação por meio de recursos computacionais, Dissertação de Mestrado, Instituto Federal de Educacao, Ciencia e Tecnologia do Estado do Ceara, Fortaleza, 2021.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Available at: https://oeis.org/. Accessed March 21, 2026.
T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations: A Problem-Based Approach, Birkhäuser, New York, 2010.
T. Koshy, Pell and Pell–Lucas numbers with applications, Springer, New York, 2014.
J. P. Robertson, Solving the generalized Pell equation $x^2-Dy^2=N$, 2004. Available at: https://www.scribd.com/document/92091818/Solving-the-Generalized-Pell-Equation. Accessed March 21, 2026.
K. M. Simons, Pell’s equations: A handshake between the ancient and the modern, Honors Thesis, Texas Christian University, Fort Worth, TX, 2022.
A. K. Pandey, B. K. Sharma, Arithmetic progressions in the values of quadratic polynomials and some figurate numbers, Palestine J. Math., 13(Special Issue II) (2024), 1–8.
F. E. B. Martinez et al., Teoria dos Numeros: Um Passeio com Primos e Outros Numeros Familiares Pelo Mundo Inteiro, 3rd ed., IMPA, Rio de Janeiro, 2013.
C. H. King, Some properties of the Fibonacci numbers, Master’s Thesis, San Jose State College, 1960.
H. W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart., 19(3) (1981), 250–257. https://doi.org/10.1080/00150517.1981.12430088
D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20(1) (1982), 73–76. https://doi.org/10.1080/00150517.1982.12430034
Y. Soykan, Generalized Guglielmo numbers: an investigation of properties of triangular, oblong and pentagonal numbers via their third order linear recurrence relations, Earthline J. Math. Sci., 9(1) (2022), 1–39. https://doi.org/10.34198/ejms.9122.139

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Francisco Hedyleno Coelho Bezerra ^*
0000-0001-5961-4857
Brazil

Francisco Regis Vieira Alves
0000-0003-3710-1561
Brazil

Eudes Antonio Costa
0000-0001-6684-9961
Brazil

Early Pub Date

October 27, 2026

Publication Date

-

Submission Date

August 18, 2026

Acceptance Date

October 7, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.33434/cams.2016687

IZ

https://izlik.org/JA48MF42NC

APA

Bezerra, F. H. C., Alves, F. R. V., & Costa, E. A. (2026). Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular. Communications in Advanced Mathematical Sciences, Advanced Online Publication, 157-170. https://doi.org/10.33434/cams.2016687

AMA

1.Bezerra FHC, Alves FRV, Costa EA. Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular. Communications in Advanced Mathematical Sciences. 2026;(Advanced Online Publication):157-170. doi:10.33434/cams.2016687

Chicago

Bezerra, Francisco Hedyleno Coelho, Francisco Regis Vieira Alves, and Eudes Antonio Costa. 2026. “Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular”. Communications in Advanced Mathematical Sciences, no. Advanced Online Publication: 157-70. https://doi.org/10.33434/cams.2016687.

EndNote

Bezerra FHC, Alves FRV, Costa EA (October 27, 2026) Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular. Communications in Advanced Mathematical Sciences Advanced Online Publication 157–170.

IEEE

[1]F. H. C. Bezerra, F. R. V. Alves, and E. A. Costa, “Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular”, Communications in Advanced Mathematical Sciences, no. Advanced Online Publication, pp. 157–170, Oct. 2026, doi: 10.33434/cams.2016687.

ISNAD

Bezerra, Francisco Hedyleno Coelho - Alves, Francisco Regis Vieira - Costa, Eudes Antonio. “Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular”. Communications in Advanced Mathematical Sciences. Advanced Online Publication (October 27, 2026): 157-170. https://doi.org/10.33434/cams.2016687.

JAMA

1.Bezerra FHC, Alves FRV, Costa EA. Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular. Communications in Advanced Mathematical Sciences. 2026;:157–170.

MLA

Bezerra, Francisco Hedyleno Coelho, et al. “Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular”. Communications in Advanced Mathematical Sciences, no. Advanced Online Publication, Oct. 2026, pp. 157-70, doi:10.33434/cams.2016687.

Vancouver

1.Francisco Hedyleno Coelho Bezerra, Francisco Regis Vieira Alves, Eudes Antonio Costa. Some Properties of the Multi-Figurate Numbers That Are Simultaneously Triangular and Centered Triangular. Communications in Advanced Mathematical Sciences. 2026 Oct. 27;(Advanced Online Publication):157-70. doi:10.33434/cams.2016687

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

download?token=eyJhdXRoX3JvbGVzIjpbXSwiZW5kcG9pbnQiOiJqb3VybmFsIiwib3JpZ2luYWxuYW1lIjoibG9nby5wbmciLCJwYXRoIjoiOGFjMy9kOTVhLzhjMDcvNmFiYTgyMWRhMGUwNTUuNDUyNDk1MDcucG5nIiwiZXhwIjoxNzkwNjExNTAyLCJub25jZSI6IjYyMDFhNzJmNDJhNDA2YjQyNDhlYjQ5MDdiNzA3YTk2In0.RNPNpKxnLdnaZEnRx9QrARMYjxpUmT227sFzEwIIN34

Technical Support by PiCODE Software

https://www.picodesoftware.com