##
**
Affinely Self-Generating Sets and Morphisms
**

###
David Garth and Adam Gouge

Division of Mathematics and Computer Science

Truman State University

Kirksville, MO 63501

USA

**Abstract:**

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 2*x* and 4*x*-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,
dvi,
ps,
latex
**

(Concerned with sequences
A000201
A001333
A003754
A003796
A028859
A032924
and
A052499
.)

Received August 4 2006;
revised versions received September 14 2006; October 19 2006.
Published in *Journal of Integer Sequences* December 30 2006.

Return to
**Journal of Integer Sequences home page**