Linear recurrence sequence associated to rays of negatively extended Pascal triangle

Hacène Belbachir, Abdelkader Bouyakoub and Fariza Krim
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 129–142
DOI: 10.7546/nntdm.2022.28.1.129-142
Full paper (PDF, 247 Kb)

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Authors and affiliations

Hacène Belbachir
USTHB, Faculty of Mathematics, RECITS Laboratory,
Po. Box 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

Abdelkader Bouyakoub
Oran1 University, Faculty of Sciences, GEAN Laboratory
Po. Box 1524, El Menaouer, 31000 Es Senia, Oran, Algeria

Fariza Krim
USTHB, Faculty of Mathematics, RECITS Laboratory,
Po. Box 32 El Alia, 16111, Algiers, Algeria
Oran1 University, Faculty of Sciences, GEAN Laboratory
Po. Box 1524, El Menaouer, 31000 Es Senia, Oran, Algeria

Abstract

We consider the extension of generalized arithmetic triangle to negative values of rows and we describe the recurrence relation associated to the sum of diagonal elements laying along finite rays. We also give the corresponding generating function. We conclude by an application to Fibonacci numbers and Morgan-Voyce polynomials with negative subscripts.

Keywords

• Binomial coefficient
• Linear recurrence
• Combinatorial identities
• Arithmetic triangle

2020 Mathematics Subject Classification

• 11B37
• 05A10
• 11B65
• 11B39
• 05A15

References

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Manuscript history

• Received: 3 February 2021
• Revised: 1 March 2022
• Accepted: 9 March 2022
• Online First: 22 March 2022