On the g-ary expansions of Apéry, Motzkin, Schröder and other combinatorial numbers

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Title: On the g-ary expansions of Apéry, Motzkin, Schröder and other combinatorial numbers
Related: Annals of combinatorics, Vol. 14, No. 4, (2010), p.507-524
DOI: 10.1007/s00026-011-0074-9
Publisher: Birkhäuser
Date: 2010
Author/Creator: Luca, Florian
Author/Creator: Shparlinski, Igor E
Description: Let g ≥ 2 be an integer and let (u n ) n≥1 be a sequence of integers which satisfies a relation u n+1 = h(n)u n 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 u n 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)u n with two nonconstant rational functions h1(X), h2(X)ϵ Q [X]. This class includes the Apéry, Delannoy, Motzkin, and Schröder numbers.
Description: 18 page(s)
Subject Keyword: 010100 Pure Mathematics
Subject Keyword: Apéry numbers
Subject Keyword: Motzkin numbers
Subject Keyword: Schröder numbers
Subject Keyword: representations in integer bases of special numbers
Subject Keyword: applications to S-unit equations
Resource Type: journal article
Organisation: Macquarie University. Department of Computing

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