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

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## Details

### 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

^{m}w_{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

### References

- A.T. Benjamin and J.J. Quinn.
*Proofs That Really Count: The Art of Combinatorial Proof*. MAA, Washington, 2003. - J.B. Gil, M.D. Weiner and C Zara. Complete Padovan Sequences in Finite Fields.
*The Fibonacci Quarterly*. 45 (2007): 64-75. - M. Hall. Divisibility Sequences of Third Order.
*American Journal of Mathematics.*58 (1936) 577-584. - V.E. Hoggatt, Jr and M. Bicknell-Johnson. Fibonacci Convolution Sequences.
*The Fibonacci Quarterly.*15 (1977): 117-122. - A.F. Horadam. Fibonacci Number Triples.
*American Mathematical Monthly.*68 (1961): 751-753. - J. Riordan. Generating Functions for Powers of Fibonacci Numbers.
*Duke Mathematical Journal*. 29 (1962): 5-12. - A.G. Shannon. Fibonacci Analogs of the Classical Polynomials.
*Mathematics Magazine.*48 (1975): 123-130. - A.G. Shannon, P.G. Anderson and A.F. Horadam. Properties of Cordonnier, Perrin and van der Laan Numbers.
*International Journal of Mathematical Education in Science and Technology*. 37.7 (2006): 825-831. - A.G Shannon and Leon Bernstein. The Jacobi-Perron Algorithm and the Algebra of Recursive Sequences.
*Bulletin of the Australian Mathematical Society.*8 (1973): 261-277. - A.G. Shannon and A.F. Horadam. Generating Functions for Powers of Third Order Recurrence Sequences.
*Duke Mathematical Journal*. 38 (1971): 791-794. - A.G. Shannon and A.F. Horadam. Generalized Pell Numbers and Polynomials. In Fredric T. Howard (ed.),
*Applications of Fibonacci Numbers, Volume 9*. Dordrecht/Boston/London: Kluwer, 2004, pp.213-224. - A.G. Shannon, A.F. Horadam, P.G. Anderson. The Auxiliary Equation Associated with the Plastic Number.
*Notes on Number Theory and Discrete Mathematics*. 12.1, (2006): 1-12. - G.J. Tee. Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetics.
*Australian Mathematical Society Gazette.*30 (2003): 267-276. - D. Terrana and H. Chen. Generating Functions for the Powers of Fibonacci Sequences.
*International Journal of Mathematical Education in Science and Technology*. 38.4 (2007): 531-537.

## Related papers

## Cite this paper

APAShannon, A. G., & Wong, C. K. (2008). Some properties of generalized third order Pell numbers. Notes on Number Theory and Discrete Mathematics, 14(4), 16-24.

ChicagoLeyendekkers, JV, and AG Shannon. “Some Properties of Generalized Third Order Pell Numbers.” Notes on Number Theory and Discrete Mathematics 14, no. 4(2008): 16-24.

MLALeyendekkers, JV, and AG Shannon. “Some Properties of Generalized Third Order Pell Numbers.” Notes on Number Theory and Discrete Mathematics 14.4 (2008): 16-24. Print.