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"Fibonacci numbers"
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On b-concatenations of two k-generalized Fibonacci numbers
Let
k
≥
2
be an integer. One of the generalization of the classical Fibonacci sequence is defined by the recurrence relation
F
n
(
k
)
=
F
n
-
1
(
k
)
+
⋯
+
F
n
-
k
(
k
)
for all
n
≥
2
with the initial values
F
i
(
k
)
=
0
for
i
=
2
-
k
,
…
,
0
and
F
1
(
k
)
=
1
.
F
n
(
k
)
is an order
k
generalization of the Fibonacci sequence and it is called
k
-generalized Fibonacci sequence or shortly
k
-Fibonacci sequence. Banks and Luca [7], among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem in more general manner by taking into account the concatenations of two terms of the same sequence in base
b
≥
2
. First, we show that there exists only finitely many such concatenations for each
k
≥
2
and
b
≥
2
. Next, we completely determine all these concatenations for all
k
≥
2
and
2
≤
b
≤
10
.
Journal Article
Growing patterns : Fibonacci numbers in nature
What's the biggest mathematical mystery in nature? Fibonacci numbers! The pattern creeps up in the most unexpected places. It's clear that math holds secrets to nature and that nature holds secret numbers.
Book
Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices
In this study, the coding theory defined for k-order Gaussian Fibonacci polynomials is rearranged by taking $ x = 1 $ $ {Q_k}, {R_k} $ $ E_n^{(k)} $ $ k = 2 $ $ k $ $ k = 2 $ $ k $
Journal Article
The man of numbers : Fibonacci's arithmetic revolution
\"The untold story of Leonardo of Pisa, the medieval mathematician who introduced Arabic numbers to the West and helped launch the modern era.\"--Page [2] of dust jacket.
Book
Generalized Cassini identities via the generalized Fibonacci fundamental system. Applications
In this paper we explore the generalized Cassini identities for the weighted generalized Fibonacci sequences, through the associated generalized Fibonacci fundamental system. Some algebraic, combinatoric and analytic properties of these identities are established. Applications to generalized Fibonacci and Pell numbers are provided, and some special cases are studied.
Journal Article
Rabbits, rabbits everywhere : a Fibonacci tale
Rapidly multiplying rabbits are taking over the village of Chee, and soon there are so many that even the Pied Piper cannot get rid of them, but a girl named Amanda discovers a pattern that leads to a way to make the rabbits leave.
Book
Generalized q-Fibonacci Sequence in Plant Morphology using q-Difference Equation
In this article, as a first step, we introduce the generalized +\" order q-difference equation using the '\" order q-difference operator. Then the generalized ti\" order Fibonacci sequence is obtained from the 1\" order q-difference equation. Also we derive certain theorems and results with Fibonacci numbers highlighting the application of principle of generalized Fibonacci sequence and the corresponding ratios. This work concludes with an application of the main result, i.e., the angle of rotation of formation of new petals, leaves and branches in flowers, plants and trees respectively can be found by changing the values of a; 5 in the generalized 1? order Fibonacci sequence. In order to substantiate our findings, we have provided sufficient examples verified by MATLAB wherever necessary.
Journal Article
Wild Fibonacci : nature's secret code revealed
Discover the fibonacci sequence as it appears in nature, from the curves of a sundial shell, to a parrot's beak, a hawk's talon, a ram's horn, and even human teeth!
Book
Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals
The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2F1(z) are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.
Journal Article