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Pell Walks

Published online by Cambridge University Press:  23 January 2015

Thomas Koshy
Thomas Koshy*
Affiliation:
Framingham State University, Framingham, MA 01701-9101, USA
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Extract

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.

Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:

where n ≥ 3. They can also be defined by Binet-like formulas:

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 538 , March 2013 , pp. 27 - 35
Copyright
Copyright © The Mathematical Association 2013

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References

1. Benjamin, A. T. et al, Counting on Chebyshev polynomials, Mathematics Magazine 82 (2009) pp. 117–126.CrossRefGoogle Scholar
2. Nkwanta, A. and Shapiro, L. W., Pell Walks and Riordan Matrices, Fibonacci Quarterly 43 (2005) pp. 170–180.Google Scholar
3. Stanley, R. P., Enumerative combinatorics, Vol. 1, Cambridge University Press, New York (2002).Google Scholar
4. Bicknell, M., A primer on the Pell sequence and related sequences, Fibonacci Quarterly 13 (1975) pp. 345–349.Google Scholar
5. Koshy, T., Pell numbers: A Fibonacci-like treasure for creative exploration, Mathematics Teacher 104 (2011) pp. 550–555.Google Scholar
6. Sulanke, R., Objects counted by the central Delannoy numbers, Journal of Integer Sequences 6 (2003), article 03.1.5.Google Scholar
7. Comtet, L., Advanced Combinatorics, D. Reidel, Dordrecht (1974).CrossRefGoogle Scholar