On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers

Florian Luca*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Citations (Scopus)


Let g ≥ 2 be an integer and let (un)n≥1 be a sequence of integers which satisfies a relation un+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 un+2 = h1(n)un+1 + h2(n)un with two nonconstant rational functions h2(X),h2(X) ∈ Q[X]. This class includes the Apéry, Delannoy, Motzkin, and Schröder numbers.

Original languageEnglish
Pages (from-to)507-524
Number of pages18
JournalAnnals of Combinatorics
Issue number4
Publication statusPublished - Dec 2010


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