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

On the Recognizability of Self-Generating Sets

Tomi Kärki
University of Turku
Department of Mathematics
FI-20014 Turku

Anne Lacroix and Michel Rigo
University of Liège
Department of Mathematics
Grande Traverse 12 (B37)
B-4000 Liège


Let $I$ be a finite set of integers and $F$ be a finite set of maps of the form $n\mapsto k_i\, n+\ell_i$ with integer coefficients. For an integer base $k\ge 2$, we study the $k$-recognizability of the minimal set $X$ of integers containing $I$ and satisfying $\varphi(X)\subseteq X$ for all $\varphi\in F$. We answer an open problem of Garth and Gouge by showing that $X$ is $k$-recognizable when the multiplicative constants $k_i$ are all powers of $k$ and additive constants $\ell_i$ are chosen freely. Moreover, solving a conjecture of Allouche, Shallit and Skordev, we prove under some technical conditions that if two of the constants $k_i$ are multiplicatively independent, then $X$ is not $k$-recognizable for any $k\ge 2$.

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(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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