Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.5

Free Fibonacci Sequences

Brandon Avila and Tanya Khovanova
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139


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.

We also introduce ten new sequences. Most of the new sequences are also related to Fibonacci numbers proper, not just free Fibonacci numbers.

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(Concerned with sequences A000032 A000045 A000057 A000285 A001175 A001177 A001602 A015134 A015135 A060305 A064362 A064414 A065156 A078414 A214684 A224382 A230359 A230457 A232357 A232656 A232658 A232666 A233246 A233248 A233525 A233526.)

Received March 28 2014; revised versions received July 29 2014; August 1 2014; August 4 2014. Published in Journal of Integer Sequences, August 5 2014.

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