Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence
School of Mathematics and Statistics
Victoria University of Wellington
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.
Full version: pdf,
(Concerned with sequences
Received June 18 2016; revised version received October 8 2016.
Published in Journal of Integer Sequences,
October 10 2016.
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