Notes on Fibonacci Partitions

Details Notes on Fibonacci Partitions Notes on Fibonacci Partitions

2003-07-10

arxiv.org/abs/math/0307150v9

Mathematics

Number Theory

17 pages, 1 figure

Scientific paper

Let $f_1=1,f_2=2,f_3=3,f_4=5,...$ be the sequence of Fibonacci numbers. It is well known that for any natural $n$ there is a unique expression $n=f_{i_1}+f_{i_2}+...+f_{i_q}$ such that $i_{a+1}-i_a2$ for $a=1,2,...,q-1$ (Zeckendorf Theorem). By means of it we find an explicit formula for the quantity $F_h(n)$ of partitions of $n$ with $h$ summands, all parts of them are the distinct Fibonacci numbers. This formula is used for an investigation of the functions $F(n)=\sum_{h=1}^\infty F_h(n)$ and $\chi(n)=\sum_{h=1}^\infty (-1)^hF_h(n)$. They are interpreted by means of the representations of rational numbers as some continues fractions. Using this approach we define a canonical action of monoid $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{N}$ (see the text for the notations) on the set of natural numbers, the set orbits of that is also a monoid, freely generated by the set $\qz$, and such that F(n) is invariant under this action. A fundamental domain of the action is found and the following results are established: the formula $\chi(n)=0,\pm 1$, a theorem on "Fibonacci random distribution" of $n$ with $F(n)=k$, the estimation $F(n)\sqrt{n+1}$, and it is shown that $\lim_{N\to\infty}(\chi^2(1)+...+\chi^2(N))/N=0$. In addition, an algorithm to find a minimal $n$ with $F(n)=k$ is provided.

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