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.
Full version: pdf,
(Concerned with sequences
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.
Journal of Integer Sequences home page