Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5

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,    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