Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence
Journal of integer sequences, Tome 19 (2016) no. 8
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.
Keywords:
cyclic composition, Euler's totient function, generalized Lucas number, generating function
@article{JIS_2016__19_8_a7,
author = {Hadjicostas, Petros},
title = {Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence},
journal = {Journal of integer sequences},
year = {2016},
volume = {19},
number = {8},
zbl = {1383.05023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_8_a7/}
}
Hadjicostas, Petros. Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence. Journal of integer sequences, Tome 19 (2016) no. 8. http://geodesic.mathdoc.fr/item/JIS_2016__19_8_a7/