The Farey sequence is an old and famous set of fractions associated with the integers. We here show that if we form a Farey sequence of Fibonacci Numbers, the properties of the Farey sequence are remarkably preserved (see [2]). In fact we find that with the new sequence we are able to observe and identify "points of symmetry/' "intervals," "generating fractions" and "stages." The paper is divid...

We know that ( ) mod n F p forms a periodic sequence (vide Theorem 4). Let ( ) h p denote the length of the sequence. Let p be a prime such that: { } ( ) 2,3 mod 5 p ≡ a sufficient and necessary condition to ensure that ( ) 2 2 h p p + . We shall denote this group 1 G F . Let { } 1, 2 , , k D d d d = be the non-empty set of k divisors of 2 2 p + . Then for ( ) [ ] 1 min G i F h p d = such that ...

Image scrambling is used to make images visually unrecognizable such that unauthorized users have difficulty decoding the scrambled image to access the original image. This article presents two new image scrambling algorithms based on Fibonacci p-code, a parametric sequence. The first algorithm works in spatial domain and the second in frequency domain (including JPEG domain). A parameter, p, i...

Complex biological patterns are often governed by simple mathematical rules. A favourite botanical example is the apparent relationship between phyllotaxis (i.e. the arrangements of leaf homologues such as foliage leaves and floral organs on shoot axes) and the intriguing Fibonacci number sequence (1, 2, 3, 5, 8, 13 . . .). It is frequently alleged that leaf primordia adopt Fibonacci-related pa...

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers. This theorem induces a binary numeration system for the positive integers known as Fibonacci coding. Fibonacci code is a variable-length prefix code that is robust against insertion and deletion errors and is useful in data transmission and data compression. In this pape...

the origin the concept of lz compexity is in information science. here we use this notion to characterize chaotic dynamical systems. we make contact with the usual characteristics of chaos, such as lyapunov exponent and k-entropy. it is shown that for a two-dimensional system lz complexity is as powerful as other characteristics. we also apply lz complexity to the study of the quasiperiodic fib...

The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, we provide various approximation results concerning the classical Korovkin theorem via Fibonacci type statistical convergence.

Consider sequences of integers {Un}TM=0 defined by U„ = aUn_x +bUn_2 for all integers n > 2, where U0 = 0, C/j = 1, a and b are given integers. We call these sequences generalized Fibonacci sequences with parameters a and b. In the case where a = b = 1, the sequence {Un}TM=0 is called the Fibonacci sequence, and we denote its terms by F0,Fl9.... The polynomial f(x) = x-ax-h with discriminant D ...

We bridge mathematical number theory with that of optimal control and show that a generalised Fibonacci sequence enters the control function of finite horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacc...