Some properties of generalized third order Pell numbers

A. G. Shannon and C. K. Wong
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 14, 2008, Number 4, Pages 16–24
Full paper (PDF, 59 Kb)

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Authors and affiliations

A. G. Shannon
Warrane College, University of New South Wales
PO Box 123, NSW 1465, Australia

C. K. Wong
Warrane College, University of New South Wales
PO Box 123, NSW 1465, Australia

Abstract

This paper considers some properties of the third order recursive sequence defined by the linear recurrence relation w_m,n = 2^mw_{m, n−2} + w_{m, n−3}, n ≥ 3, m = 0, 1, 2, with appropriate initial conditions. The present work follows on from the case m = 0 (Shannon et al). Relationships with the well-known sequences of Fibonacci, Lucas and Pell are developed. The motivation for the study was to find analogous results to some of the second order classic identities such as, for example, Simson’s identity and Horadam’s Fibonacci number triples.

Keywords

• Fibonacci
• Lucas
• Pell numbers
• Simson
• Convolutions
• Generating functions

AMS Classification

• 11B37
• 11B39

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