Related papers: Partial Sums of the Fibonacci Sequence

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

In this paper we compute the Frobenius number of certain {\em Fibonacci numerical semigroups}, that is, numerical semigroups generated by a set of Fibonacci numbers, in terms of Fibonacci numbers.

The close relationship among the polynomial functions and Fibonacci numerical sequences is shown in this paper. These numerical sequences are defined by the recurrence equation $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}\alpha_j x_{k +…

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers.…

One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.

In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.

In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving…

We study the extended Frobenius problem for sequences of the form $\{f_a+f_n\}_{n\in\mathbb{N}}$, where $\{f_n\}_{n\in\mathbb{N}}$ is the Fibonacci sequence and $f_a$ is a Fibonacci number. As a consequence, we show that the family of…

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…

In this paper we study some properties of Fibonacci-sum set-graphs. The aforesaid graphs are an extension of the notion of Fibonacci-sum graphs to the notion of set-graphs. The colouring of Fibonacci-sum graphs is also discussed. A number…

The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $\sigma(a,b)=(ab,a)$. Since $\mathbb{F}$ is uniformly recurrent, each factor $\omega$ appears infinite many times in the sequence which is arranged as…

We recursively define a sequence $\{F_{n,k}\}_{n,k\in\mathbb N }$ and we prove that such sequence contains only the symbols $\{0,1\}.$ We investigate some number-theoretic properties of such sequence and of the way it can be generated. The…

Let $\left\lbrace F_{k}\right\rbrace_{k\geq0}$ be the Fibonacci sequence defined by $F_{k}=F_{F-1}+F_{k-2}$ for all $ n\geq2$ with initials $F_{0}=0\; F_{1}=1$. Let $\left\lbrace J_{n}\right\rbrace_{n\geq0}$ be the Jacobsthal sequence…

A \Def{composition} of a positive integer $n$ is a $k$-tuple $(\l_1, \l_2, \dots, \l_k) \in \Z_{> 0}^k$ such that $n = \l_1 + \l_2 + \dots + \l_k$. Our goal is to enumerate those compositions whose parts $\l_1, \l_2, \dots, \l_k$ avoid a…

In this paper, we study the linear space of all two-sided generalized Fibonacci sequences $\{F_n\}_{n \in \mathbb{Z}}$ that satisfy the recurrence equation of order $k$: $F_n = F_{n-1} + F_{n-2} + \dots + F_{n-k}$. We give two types of…

We study the complexity classes P and NP through a semigroup fP ("polynomial-time functions"), consisting of all polynomially balanced polynomial-time computable partial functions. Then P is not equal to NP iff fP is a non-regular…

In this paper, we use Abel's summation formula to evaluate several quadratic and cubic sums of the form: \[{F_N}\left( {A,B;x} \right) := \sum\limits_{n = 1}^N {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right){x^n}} ,\;x \in [ - 1,1]\]…