On Recurrences in Generalized Arithmetic Triangle-Reference-Cited by-同舟云学术

On Recurrences in Generalized Arithmetic Triangle

Published:2023-03-31 Issue:2 Volume:73 Page:305-334
ISSN:1337-2211
Container-title:Mathematica Slovaca
language:en
Short-container-title:

Author:

Belbachir Hacéne¹²,Bouyakoub Abdelkader³,Krim Fariza⁴⁵

Affiliation:

1. * USTHB, Faculty of Mathematics , RECIT Laboratory , Po. Box 32, El Alia, 16111 , Algiers , ALGERIA

2. CERIST: Research Center for Scientific and Technical Information , 05 Rue des trois fréres Aissou, Ben Aknoun , Algiers , ALGERIA , belbachir@cerist.dz

3. ** Faculty of Sciences , Oran1 University , GEAN Laboratory , Po. Box 1524, El Menaouer, 31000 Es Senia , Oran , ALGERIA

4. *** Faculty of Sciences , Oran1 University , GEAN Laboratory , Po. Box 1524, El Menaouer, 31000 Es Senia , Oran , ALGERIA

5. USTHB, Faculty of Mathematics , RECIT Laboratory , Po. Box 32, El Alia, 16111 Algiers , ALGERIA

Abstract

Abstract In the present paper, we consider the generalized arithmetic triangle called GAT which is structurally identical to Pascal’s triangle for which we keep the Pascal’s rule of addition and we replace both legs by two sequences (an ) n≥1 and (bn ) n≥1 with a 0 = b 0 = Ω. Our goal is to describe the recurrence relation associated to the sum of elements lying along a finite ray in this triangle. As consequences, we obtain some combinatorial properties and we establish that the sum of elements lying along a main rising diagonal is a convolution of generalized Fibonacci sequence and another sequence which one will determine. We also precise the corresponding generating function. Further, we establish some nice identities by using the Morgan-Voyce phenomenon. Finally, we generalize the Golden ratio.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/ms-2023-0025/pdf

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