Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Notes on Fibonacci Partitions   Let $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of
Fibonacci numbers. Let $\Phi_h(n)$ be the quantity of partitions of natural
number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf
partition of $n$ I deduce a formula for the function $\Phi(n;t):=\sum_{h\geq
1}\Phi_h(n)t^h$, and use it to analyze the functions $F(n):=\Phi(n;1)$ and
$\chi(n):=\Phi(n;-1)$. I obtain the least upper bound for $F(n)$ when
$f_i-1\<n\<f_{i+1}-1$. It implies that $F(n)\<\sqrt{n+1}$ for any natural $n$.
  I prove also that $|\chi(n)|\<1$, and
$\mathop{\lim}\limits_{N\to\infty}\frac{1}{N}
\left(\chi^2(1)+\chi^2(2)\dpt\chi^2(N)\right)=0$.
  For any $k\>2$, I define a special finite set $\mathbb{G}(k)$ of solutions of
the equation $F(n)=k$, all solutions can be easily obtained from
$\mathbb{G}(k)$. This construction uses a representation of rational numbers as
certain continued fractions and provides with a canonical identification
$\coprod_{k\>2}\mathbb{G}(k)=\G_+$, where $\G_+$ is the monoid freely generated
by the positive rational numbers $<1$.
  Let $\Psi(k)$ be the cardinality of $\mathbb{G}(k)$. I prove that, for
$i\>2k$ and $k\>2$, the interval $[f_i-1,f_{i+1}-1]$ contains exactly
$2\Psi(k)$ solutions of the equation $F(n)=k$ and offer a formula for the
Dirichlet generating function of the sequence $\Psi(k)$. I formulate
conjectures on the set of minimal solutions of the equations $F(n)=k$ as $k$
varies and pose some questions concerning such solutions.

30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
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The categories of this paper are: econ
32
The categories of this paper are: atom-ph