Geodesic
Free Fibonacci sequences
Journal of integer sequences, Tome 17 (2014) no. 8
This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given $n$, sum the previous two terms and divide them by the largest possible power of $n$. The behavior of such sequences depends on $n$. We analyze these sequences for small $n$: 2, 3, 4, and 5. Surprisingly, these behaviors are very different. We also present theorems regarding any $n$. Many statements about these sequences may be difficult or even impossible to prove, but they can be supported by probabilistic arguments. We have plenty of those in this paper.
Classification : 11B39, 11B50
Keywords: Fibonacci number, Lucas number, divisibility, entry point 
@article{JIS_2014__17_8_a5,
     author = {Avila,  Brandon and Khovanova,  Tanya},
     title = {Free {Fibonacci} sequences},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {8},
     zbl = {1298.11013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_8_a5/}
}
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Avila,  Brandon; Khovanova,  Tanya. Free Fibonacci sequences. Journal of integer sequences, Tome 17 (2014) no. 8. http://geodesic.mathdoc.fr/item/JIS_2014__17_8_a5/