The Period and the Distribution of the Fibonacci-like Sequence Under Various Moduli

We reduce the Fibonacci sequence mod m for a natural number m, and denote it by F (mod m). We are going to introduce the properties of the period and distribution of F (mod m). That is, how frequently each residue is expected to appear within a single period. These are well known themes of the research of the Fibonacci sequence, and many remarkable facts have been discovered. After that we are going to study the properties of period and distribution of a Fibonacci-like sequence that the authors introduced in [4]. This Fibonacci-like sequence also has many interesting properties, and the authors could prove an interesting theorem in this article. Some of properties are very difficult to prove, and hence we are going to present some predictions and calculations by computers.