A note on the self-convolution of the Tribonacci sequence

Karol Gryszka
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 133–136
DOI: 10.7546/nntdm.2026.32.1.133-136
Full paper (PDF, 168 Kb)

Details

Authors and affiliations

Karol Gryszka
Institute of Mathematics, University of the National Education Commission, Krakow
Podchorążych 2, 30-084 Krakow, Poland

Abstract

We present a simple formula for the self-convolution of the Tribonacci numbers. The resulting identity is considerably simpler than that obtained in a recent publication.

Keywords

• Tribonacci numbers
• Convolution

2020 Mathematics Subject Classification

• 11B39

References

1. Dresden, G., & Wang, Y. (2021). Sums and convolutions of k-onacci and k-Lucas numbers. Integers, 21, Article ID A56.
2. Frontczak, R. (2018). Convolutions for generalized Tribonacci numbers and related results. International Journal of Mathematical Analysis, 12(7), 307–324.
3. Frontczak, R. (2018). Some Fibonacci–Lucas–Tribonacci–Lucas identities. The Fibonacci Quarterly, 56(3), 263–274.
4. Komatsu, T. (2018). Convolution identities for Tribonacci numbers. Ars Combinatoria, 136, 199–210.
5. Komatsu, T. (2019). Convolution identities for Tribonacci-type numbers with arbitrary initial values. Palestine Journal of Mathematics, 8(2), 413–417.
6. Komatsu, T., & Li, R. (2019). Convolution identities for Tetranacci numbers. Notes on Number Theory and Discrete Mathematics, 25(3), 142–169.
7. Rabinowitz, S. (1996). Algorithmic manipulation of third-order linear recurrences. The Fibonacci Quarterly, 34(5), 447–464.
8. Wilf, H. S. (1994). Generatingfunctionology. Academic Press

Manuscript history

• Received: 30 November 2025
• Revised: 19 February 2026
• Accepted: 26 February 2026
• Online First: 26 February 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).