Journal of Integer Sequences, Vol. 25 (2022), Article 22.5.3 |

Department of Mathematics

Room 121, 1984 Mathematics Road

Vancouver, BC V6T 1Z2

Canada

Richard K. Guy

Renate Scheidler

Department of Mathematics and Statistics

University of Calgary

2500 University Drive NW

Calgary, AB T2N 1N4

Canada

**Abstract:**

An *S*-necklace of length *n* is a circular arrangement of the
integers 0, 1, 2, ..., *n* – 1 such that the absolute difference
of two neighbors always belongs to *S*. Focusing in particular on the
case |*S*| = 2, we prove that, subject to certain conditions on
the two elements in *S*, the number of *S*-necklaces obeys a
linear homogeneous recurrence relation. We give an algorithm for computing
the corresponding generating function and compute generating functions
and explicit recurrence relations for several small sets *S*. Our
methods extend to sets *S* of any size.

(Concerned with sequences A003520 A017899 A079977.)

Received February 22 2022; revised version received June 12 2022.
Published in *Journal of Integer Sequences*,
June 13 2022.

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