Related papers: Notes on Fibonacci Partitions

We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = \lambda F_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |\lambda \widetilde F_{n+1} \pm \widetilde F_{n}|$…

Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into $n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set $F\subset G$…

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…

In this paper, harkening back to ideas of Hardy and Ramanujan, Mahler and de Bruijn, with the addition of more recent results on the Fibonacci Dirichlet series, we determine the asymptotic number of ways $p_F(n)$ to write an integer as the…

We study $B(n;k)$, the number of ways of writing $n$ as a sum or difference of the first $k$ Fibonacci numbers. We show that $B(0;k)$ satisfies the Tribonacci-like recurrence $B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2)$ and that $B(n;k)$ satisfies a…

Let $F(X)= \prod_{i=1}^k(a_iX+b_i)$ be a polynomial with $a_i, b_i$ being integers. Suppose the discriminant of $F$ is non-zero and $F$ is admissible. Given any natural number $N$, let $S(F,N)$ denotes those integers less than or equal to…

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty,…

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed…

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of…

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution…

An approximate formula for the partitions of Goldbach's Conjecture is derived using Prime Number Theorem and a heuristic probabilistic approach. A strong form of Goldbach's conjecture follows in the form of a lower bounding function for the…

We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.

Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on…

Zeckendorf proved that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing an integer $n$ into a 2-player game, using the…

Let $F_n$ denote the $n^{th}$ Fibonacci number relative to the initial conditions $F_0=0$ and $F_1=1$. Bach, Paudyal, and Remmel introduced Fibonacci analogues of the Stirling numbers called Fibo-Stirling numbers of the first and second…

Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from $1, 2, 3, 5,\ldots$. This has been generalized by many authors, in particular to constant…

Recently, Chu studied some properties of the partial sums of the sequence $P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and $(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial interpretation. We generalize…

Let $f_1,\dots,f_k\in\mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant…

Let $\{G_n\}$ be a periodic sequence of integers modulo $m$ and let $\{SG_n\}$ be the partial sum sequence defined by $SG_n:= \sum_{k=0}^nG_k $ (mod $m$). We give a formula for the period of $\{SG_n\}$. We also show that for a generalized…