• Journal for History of Mathematics
• /
• v.21 no.4
• /
• pp.87-104
• /
• 2008

In this paper we investigate several properties and characteristics of the generalized Fibonacci sequence $\{g_n\}$={a, b, a+b, a+2b, 2a+3b, 3a+5b,...}. This concept is a generalization of the famous Fibonacci sequence. In particular we find the identities of sums and the nth term $g_n$ in detail. Also we find the generalizations of the Catalan's identity and A. Tagiuri's identity about the Fibonacci sequence, and investigate the relation between $g_n$ and Pascal's triangle, and how fast $g_n$ increases. Furthermore, we show that $g_n$ and $g_{n+1}$ are relatively prime if a b are relatively prime, and that the sequence $\{\frac{g_{n+1}}{g_n}\}$ of the ratios of consecutive terms converges to the golden ratio $\frac{1+\sqrt5}2$.

• Journal of the Korean Mathematical Society
• /
• v.42 no.4
• /
• pp.761-771
• /
• 2005

In this paper we present another characterization of (${\pm}1$)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence $x\;{\in}\;R^{\infty}$ is (-1)-invariant(l-invariant resp.) if and only if $D[_x^0]$ is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where $$D=diag((-1)^0,\; (-1)^1,\;(-1)^2,{\ldots})$$.

• School Mathematics
• /
• v.12 no.4
• /
• pp.619-638
• /
• 2010

In this paper, we listed and reviewed some properties on polygonal numbers, pyramidal numbers and Pascal's triangle, and Fibonacci sequence. We discussed that the properties of gnomonic numbers, polygonal numbers and pyramidal numbers are explained integratively by introducing the generalized k-dimensional pyramidal numbers. And we also discussed that the properties of those numbers and relationships among generalized k-dimensional pyramidal numbers, Pascal's triangle and Fibonacci sequence are suitable for teaching and learning of mathematical reasoning and connections.

• The Korean Journal of Applied Statistics
• /
• v.22 no.5
• /
• pp.1103-1113
• /
• 2009

We know that the first digits sequence of fibonacci numbers obey Benford's law. For the sequence in which the first two numbers are the arbitrary integers and the recurrence relation $a_{n+2}=a_{n+1}+a_n$ is satisfied, we can find that the first digits sequence of this sequence obey Benford's law. Also, we can find the stucture of the first digits sequence of this sequence with the exploratory data analysis tools.

• The Pure and Applied Mathematics
• /
• v.20 no.3
• /
• pp.137-148
• /
• 2013

In this paper, we study the Lie-generalized Fibonacci sequence and the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We derive several interesting properties of the Lie-Fibonacci sequence and relationship between them. We also give a couple of sufficient conditions for the existence of the integral points on the hyperbola $\mathfrak{h}^a:x^2-axy+y^2=1$ and $\mathfrak{h}_k:x^2-axy+y^2=-k$ ($k{\in}\mathbb{Z}_{>0}$). To list all the integral points on that hyperbola, we find the number of elements of ${\Omega}_k$.

• Journal of the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.135-151
• /
• 2005

In this paper, we obtain some properties of the sequences $U^{q}_n\;and\;V^{q}_n$ introduced in [6]. We find polynomial representations and formulas of them. For q = 5, $U^{5}_n$ is the Fibonacci sequence $F_n\;and\;V^{5}_n$ is the Lucas sequence $L_n$.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.4
• /
• pp.1033-1048
• /
• 2020

We synthesize the recent work done on conditionally defined Lucas and Fibonacci numbers, tying together various definitions and results generalizing the linear recurrence relation. Allowing for any initial conditions, we determine the generating function and a Binet-like formula for the general sequence, in both the positive and negative directions, as well as relations among various sequence pairs. We also determine conditions for periodicity of these sequences and graph some recurrent figures in Python.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.511-522
• /
• 2017

We explicitly solve the diophantine equations of the form $$A_{n_1}A_{n_2}{\cdots}A_{n_k}{\pm}1=B^2_m$$, where $(A_n)_{n{\geq}0}$ and $(B_m)_{m{\geq}0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques [9] and L. Szalay [13] concerning a variant of Brocard-Ramanujan equation.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.4
• /
• pp.327-334
• /
• 2021

In this paper, we find all the prime numbers p that satisfy the following statement. If a positive integer k is a divisor of p - 1, then there is a sequence consisting of all k-th power residues modulo p, satisfying the recurrence equation of the Fibonacci sequence modulo p.

• Proceedings of the IEEK Conference
• /
• 2001.06c
• /
• pp.63-66
• /
• 2001

In this paper, We propose the new interconnection network which is designed to edge numbering labeling using postorder traversal which can reduce diameter when it has same node number with Hypercube, which can reduce total node numbers considering node degree and diameter on Fibonacci trees and its jump sequence of circulant is Fibonacci numbers. It has a simple (shortest oath)routing algorithm, diameter, node degree. It has a spaning subtree which is Fibonacci tree and it is embedded to Fibonacci tree. It is compared with Hypercube. We improve diameter compared with Hypercube.