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**
Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence
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Petros Hadjicostas

School of Mathematics and Statistics

Victoria University of Wellington

Wellington 6140

New Zealand

**Abstract:**

A linear composition of a positive integer *n* is a finite sequence of
positive integers (called parts) whose sum equals *n*. A cyclic
composition of *n* is an equivalent class of all linear compositions of
*n* that can be obtained from each other by a cyclic shift. In this paper,
we enumerate the cyclic compositions of *n* that avoid an increasing
arithmetic sequence of positive integers. In the case where all
multiples of a positive integer *r* are avoided, we show that the number
of cyclic compositions of *n* with this property equals to or is one less
than the number of cyclic zero-one sequences of length *n* that do not
contain *r* consecutive ones. In addition, we show that this number is
related to the *r*-step Lucas numbers.

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(Concerned with sequences
A000032
A000073
A000078
A000358
A001350
A001590
A001631
A001644
A008965
A032189
A037306
A073817
A093305.)

Received June 18 2016; revised version received October 8 2016.
Published in *Journal of Integer Sequences*,
October 10 2016.

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