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.
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Benjamin, A.T., Plott, S.S., & Sellers, J.A. (2008). Tiling proofs of recent sum identities involving Pell numbers. Annals of Combinatorics, 12(3): 271-278. DOI: 10.1007/s00026-008-0350-5.