Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2 |

University of Turku

Department of Mathematics

FI-20014 Turku

Finland

Anne Lacroix and Michel Rigo

University of Liège

Department of Mathematics

Grande Traverse 12 (B37)

B-4000 Liège

Belgium

**Abstract:**

Let be a finite set of integers and be a finite set of
maps of the form
with integer
coefficients. For an integer base , we study the
-recognizability of the minimal set of integers containing
and satisfying
for all .
We answer an open problem of Garth and Gouge by showing that is -recognizable when the multiplicative constants are all powers of and additive constants are chosen freely.
Moreover, solving a conjecture of Allouche, Shallit and Skordev, we
prove under some technical conditions that if two of the constants
are multiplicatively independent, then is not
-recognizable for any .

(Concerned with sequences A000045 A000201 A001950 A003754 A003849 A052499.)

Received November 16 2009;
revised versions received January 21 2010.
Published in *Journal of Integer Sequences*, January 27 2010.

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