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Combinatorial proofs of various \(q\)-Pell identities via tilings. (English) Zbl 1233.05037

Summary: Recently, A.T. Benjamin, S.S. Plott, and J.A. Sellers [Ann. Comb. 12, No.3, 271–278 (2008; Zbl 1169.05305)] proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove \(q\)-analogues of several Pell identities via weighted tilings.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05B45 Combinatorial aspects of tessellation and tiling problems
05A15 Exact enumeration problems, generating functions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations

Citations:

Zbl 1169.05305
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References:

[1] Andrews G.E.: The Theory of Partitions. Addison-Wesley, Reading (1976) · Zbl 0371.10001
[2] Benjamin A.T., Plott S.P., Sellers J.A.: Tiling proofs of recent sum identities involving Pell numbers. Ann. Combin. 12(3), 271–278 (2008) · Zbl 1169.05305 · doi:10.1007/s00026-008-0350-5
[3] Santos J.P.O., Sills A.V.: q-Pell sequences and two identities of V. A. Lebesgue. Discrete Math. 257, 125–142 (2002) · Zbl 1007.05017 · doi:10.1016/S0012-365X(01)00475-7
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