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Tiling Proofs of Recent Sum Identities Involving Pell Numbers
Arthur T. Benjamin1, Sean S. Plott1, and James A. Sellers2
1Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
{benjamin, splott}@hmc.edu
2Department of Mathematics, Penn State University, University Park, PA 16802, USA
sellersj@math.psu.edu
Annals of Combinatorics 12 (3) pp.271-278 September, 2008
AMS Subject Classification: 05A19
Abstract:
In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.
Keywords: Pell numbers, combinatorial identities, tilings, NSW numbers

References:

1. A.T. Benjamin and J.J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions, Vol. 27, Mathematical Association of America, Washington, DC, 2003.

2. M. Newman, D. Shanks, and H.C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith. 38 (2) (1980) 129–-140.

3. S.F. Santana and J.L. Diaz-Barrero, Some properties of sums involving Pell numbers, Missouri J. Math. Sci. 18 (1) (2006) 33-–40.

4. N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/njas/sequences/, 2006.