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On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers
Florian Luca1 and Igor E. Shparlinski2
1Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, C.P. 58089, México
2Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Annals of Combinatorics 14 (4) pp.507-524 December, 2010
AMS Subject Classification: 05A10, 11B83, 11D45
Let g≥ 2 be an integer and let (un){n≥ 1} be a sequence of integers which satisfies a relation u{n+1} = h(n) un for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of un in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u{n+2} = h1(n) u{n+1} + h2(n) un with two nonconstant rational functions h1(X), h2(X)∈ Q[X]. This class includes the Apéry, Delannoy, Motzkin, and Schröder numbers.
Keywords:Apéry numbers, Motzkin numbers, Schröder numbers, representations in integer bases of special numbers, applications to S-unit equations


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