A. G. Shannon, A. F. Horadam and Peter G. Anderson

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 12, 2006, Number 1, Pages 1–12

**Full paper (PDF, 112 Kb)**

## Details

### Authors and affiliations

A. G. Shannon

*KvB Institute of Technology, North Sydney, 2060 &
Warrane College, University of New South Wales, Kensington,1465, Australia
*

A. F. Horadam

*The University of New England, Armidale, 2351, Australia
*

Peter G. Anderson

*Department of Computer Science, Rochester Institute of Technology, NY14623-5608
*

### Abstract

This paper looks at some of the properties of the auxiliary equation associated with the plastic number which, in turn, is related to the sequences of numbers {*P _{n}*}, {

*Q*} and {

_{n}*R*}, respectively, defined by

_{n}*P*=

_{n}*P*

_{n − 2}+

*P*

_{n − 3},

*n*> 3,

*P*

_{1}= 1,

*P*

_{2}= 1,

*P*

_{3}= 1,

*Q*=

_{n}*Q*

_{n − 2}+

*Q*

_{n − 3},

*n*> 3,

*Q*

_{1}= 0,

*Q*

_{2}= 2,

*Q*

_{3}= 3,

*R*=

_{n}*R*

_{n − 2}+

*R*

_{n − 3},

*n*> 3,

*R*

_{1}= 1,

*R*

_{2}= 0,

*R*

_{3}= 1.

The dominant root of the associated auxiliary equation is found by a contraction process related to Bernoulli’s iteration and the Jacobi–Perron Algorithm. The latter is one way of generalizing the ordinary continued fraction algorithm and an alternative way is explored which also relates to the auxiliary equations of the sequences. Various methods for reduction of the order of the cubic auxiliary equation are also considered.

### AMS Classification

- 11B37
- 12D10
- 65D15

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## Cite this paper

Shannon, A. G., Horadam, A. F., & Anderson, P. G. (2006). The auxiliary equation associated with the plastic number. *Notes on Number Theory and Discrete Mathematics*, 12(1), 1-12.