Dear editor.
We are going to submit our article "The Period and the Distribution of the Fibonacci-like Sequence Under Various Moduli".
We have published our article "Formulas for Fibonacci-like sequences produced by Pascal-like Triangles" in Undergraduate Math Journal, and our new article is based on our previous article.
We have completed this work in January of 2009. Two of us are high school students and another is in his second year at university.
A biographical sketch.
Hiroshi Matsui is now a student at Kwansei Gakuin High School, and he is going to major in chemistry at Osaka University. He wants to be a scientist.
Matsui loves to play the piano and Shogi(Japanese chess), do C++ programming and video games. Matsui is going to attend the Intel International Science and Engineering Fair representing Japan.
Masakazu Naito is now a student at Kwansei Gakuin High School. He loves mathematics and video games. He divides his time between research and hunting monsters in video games with the ratio of 1:3, but his concentration often produces remarkable result.
Naoyuki Totani is in his second year in Kwansei Gakuin University science department. He is majoring in computer science, and he loves programming. His favorite languages are Java and Python. He has been working as an instructor in Information and Communication Technology School sponsored by Japanese Government for 2 years.
He wants to be a researcher of artificial intelligence in the future.
Matsui and Naito have worked hard to find new facts using computers, and Totani has backed up the research with his knowledge of computer.
Matsui proved theorems mathematically with help form Dr.Ryohei Miyadera.
Abstract
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 article in the previous issue of Undergraduate Math Journal. 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.