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 talk about any n. Many statements about these sequences are difficult or 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.
Fibonacci Numbers and n-free Fibonacci SequencesLet us denote Fibonacci numbers by F k . We assume that F 0 = 0 and F 1 = 1. The sequence is defined by the Fibonacci recurrence: F n+1 = F n + F n−1 (See A000045). We call an integer sequence a n Fibonacci-like if it satisfies the Fibonacci recurrence: a k = a k−1 + a k−2 . A Fibonacci-like sequence is similar to the Fibonacci sequence, except it starts with any two integers. The second-famous Fibonacci-like sequence is the sequence of Lucas numbers L i that starts with L 0 = 2 and L 1 = 1: 2, 1, 3, 4, 7, 11, . . . (See A000032).An n-free Fibonacci sequence starts with any two integers: a 1 and a 2 and is defined by the recurrence a k = (a k−1 + a k−2 )/n i , where n i is the largest power of n that is a factor of a k−1 + a k−2 . To continue the tradition we call numbers in the n-free Fibonacci sequence that starts with a 0 = 0 and a 1 = 1 n-free Fibonacci numbers.In the future we will consider only sequences starting with two non-negative integers. It is not that we do not care about other starting pairs, but positive sequences cover all essential cases. Indeed, if we start with two negative numbers we can multiply the sequence by −1 and get an all-positive sequence. If we start with numbers of different signs, the sequence eventually will become the same-sign sequence.If we start with two zeros, we get an all-zero sequence. So we will consider only sequences that do not have two zeros at the beginning. Note, that a non-negative sequence can have a zero only in one of the two starting positions, never later.The n-free Fibonacci sequence coincides with the Fibonacci-like sequence with the same beginning until the first occurrence of a multiple of n in the Fibonacci-like sequence.Given a positive integer m > 1, the smallest positive index k for which n divides the k-th Fibonacci number F k is called the entry point of m and is denoted by Z(m) (see sequence A001177 of Fibonacci entry points). For example, Z(10) = 15 and the 10-free Fibonacci
