Related papers: Simson Identity of Generalized m-step Fibonacci Nu…
The Fibonacci sequence modulo $m$, which we denote $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ where $\mathcal{F}_{m,n}$ is the Fibonacci number $F_n$ modulo $m$, has been a well-studied object in mathematics since the seminal paper by…
Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers $f_n$ of the form $1f_n = f_n$ for all $n \geq 1$; $2f_n = f_{n-2} + f_{n+1}$ for all $n \geq 3$; $3f_n = f_{n-2} + f_{n+2}$ for all $n \geq 3$;…
Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping,…
In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots,…
We prove that $|Av_n(231,312,1432)|$, $|Av_n(312,321,1342)|$ $|Av_n(231,312,4321,21543)|$, and $ |Av_n(321,231,4123,21534)|$, are all equal to $F_{n+1} - 1$ where $F_n$ is the $n$-th Fibonacci number using the convention $F_0 = F_1 = 1$ and…
The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $\sigma(a,b)=(ab,a)$. In this paper, we get the explicit expressions of all squares and cubes, then we determine the number of distinct squares and cubes…
In this paper, we consider the new family of recurrence sequences of $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze…
For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for…
We derive the double recurrence $e_n = \frac{1}{2}(a_{n-1}+5b_{n-1}); f_{n} = \frac{1}{2}(a_{n-1}+b_{n-1})$ with $e_0=2;f_0=0$ for the Fibonacci numbers, leading to an extremely simple and fast implementation. Though the recurrence is…
We give alternative proof of Melham's and Howard's identities on generalised Fibonacci numbers
Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i…
If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the…
We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…
We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
In this paper, we connect two well established theories, the Fibonacci numbers and the Jordan algebras. We give a series of matrices, from literature, used to obtain recurrence relations of second-order and polynomial sequences. We also…
Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and…
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.