Geodesic
Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence
Journal of integer sequences, Tome 19 (2016) no. 8.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     author = {Hadjicostas, Petros},
     title = {Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
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     number = {8},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_8_a7/}
}
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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/