TY - JOUR
T1 - On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers
AU - Luca, Florian
AU - Shparlinski, Igor E.
PY - 2010/12
Y1 - 2010/12
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=79952699908&partnerID=8YFLogxK
U2 - 10.1007/s00026-011-0074-9
DO - 10.1007/s00026-011-0074-9
M3 - Article
AN - SCOPUS:79952699908
SN - 0218-0006
VL - 14
SP - 507
EP - 524
JO - Annals of Combinatorics
JF - Annals of Combinatorics
IS - 4
ER -