Related papers: Formula for the $n$th $k$-Generalized Fibonacci-li…

In this paper,we find all generalized Fibonacci numbers which are Narayana's cows numbers. In our proofs, we use both Baker's theory of nonzero linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method.

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Fa\`a di Bruno's formula.…

Let $ k \geq 2 $ and let $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with certain initial $ k $ terms and each term afterward is the sum of the $ k $ preceding terms. In this paper, we find all repdigits which are…

In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.

Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular…

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…

In this article we gave a recurrence to obtain the n-th prime number as function of the (n-1)-th prime number.

In this paper, a product formula for the one-dimensional $(k,a)$-generalized Fourier kernel is given for $k\geq0$, $a>0$ and $2k>a-1$, extending the special case when $a=\frac{2}{n}$, $n\in \mathbb{N}$.

In this note we generalize the trace inequality derived by [1] to the case where the number of terms of the sum (denoted by K) is arbitrary.

The random Fibonacci chain is a generalisation of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 01$ with probability $p$ and $1 \mapsto 10$ with probability $1-p$ for $0<p<1$ and where…

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{th}$ Feng-Rao number is obtained.

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…

The paper explores combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. These infinite sequences, measures the diversity of subwords in Fibonacci words, showing non-decreasing…

A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the…

The Collatz sequence for a given natural number $N$ is generated by repeatedly applying the map $N$ $\rightarrow$ $3N+1$ if $N$ is odd and $N$ $\rightarrow$ $N/2$ if $N$ is even. One elusive open problem in Mathematics is whether all such…

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…

Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erd\"os and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is…

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a)=ab$, $\sigma(b)=ac$, $\sigma(c)=a$. The prefix of $\mathbb{T}$ of length $n$ is denoted by $\mathbb{T}[1,n]$. The main result is threefold, we give: (1)…

In this paper, we introduce a variation of the factor complexity, called the $N$-factor complexity, which allows us to characterize the complexity of sequences on an infinite alphabet. We evaluate precisely the $N$-factor complexity for the…

In this paper, we first give new generalizations for third-order Horadam $\{H_{n}^{(3)}\}_{n\in \mathbb{N}}$ and generalized Tribonacci $\{h_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for classic Horadam and generalized Fibonacci numbers.…