Affinely Self-Generating Sets and Morphisms
David Garth and Adam Gouge
Division of Mathematics and Computer Science
Truman State University
Kirksville, MO 63501
Kimberling defined a self-generating set S of integers as follows.
Assume 1 is a member of S and if x is in S then 2x and 4x-1
are also in S. We study similar self-generating sets of integers
whose generating functions come from a class of affine functions for
which the coefficients of x are powers of a fixed base. We prove
that for any positive integer m the resulting sequence, reduced
modulo m, is the image of an infinite word that is the fixed point of
a morphism over a finite alphabet. We also prove that the resulting
characteristic sequence of S is the image of the fixed point of a
morphism of constant length, and is therefore automatic. We then give
several examples of self-generating sets whose expansions in a certain
base are characterized by sequences of integers with missing blocks of
digits. This expands upon earlier work by Allouche, Shallit, and
Skordev. Finally, we give another possible generalization of the
original set of Kimberling.
Full version: pdf,
(Concerned with sequences
Received August 4 2006;
revised versions received September 14 2006; October 19 2006.
Published in Journal of Integer Sequences December 30 2006.
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