Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.2

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

Petros Hadjicostas
School of Mathematics and Statistics
Victoria University of Wellington
Wellington 6140
New Zealand


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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