The aim of this paper is to introduce the cyclic-Fibonacci hybrid sequence and give some properties. By taking into account the cyclic-Fibonacci hybrid sequence modulo m, the method will be given to determine the period lengths of this sequence according to the different m values. In the final part of this paper, we study the cyclic-Fibonacci hybrid sequence in groups and then we calculate the cyclic-Fibonacci hybrid lengths of polyhedral groups (2, 2, 2), (2, n, 2) and (n, 2, 2) as applications of the results produced.

In the present paper, we aim to generalize the notion of complex-type Fibonacci sequences to complex-type cyclic Fibonacci sequences. Firstly, we define the complex-type cyclic-Fibonacci sequence and then we give miscellaneous  properties of this sequence by using the matrix method. Also, we study the complex-type cyclic-Fibonacci sequence modulo $m$. In addition, we describe the complex-type cyclic-Fibonacci sequence in a $2$-generator group and investigate that in finite groups in details. Then, as our last result, we obtain the lengths of the periods of the complex-type cyclic-Fibonacci sequences in dihedral groups $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$, $D_{6}$ and $D_{8}$ with respect to their generating sets.

This paper try to show that cling to the physical order that governs nature can be helpful in engineering application such as acoustical materials. In this regards, an innovative idea to the noise control problem for broader absorption bandwidth at low frequencies is presented in this paper. For this aim, Helmholtz resonator array (HRA) designed based on Fibonacci sequence, which is addressed “, Fibonacci based-HRA”, . First, to validate the simulations which were performed through finite element method (FEM) as a numerical method in COMSOL Multi-physics software (Version 5. 2), the results have been compared with experimental results which obtained by other researchers for two cases: (1) one unit resonator, and (2) four resonators array block with equal size. Second, to examine the performance of sound absorption of Fibonacci based-HRA, the results obtained for HRA with equal sizes of the 4, 6, 8 and 10 component resonators, compared with those obtained for Fibonacci based-HRA with the same number of the component resonators and the same overall dimensions of HRA block. The comparison showed that Fibonacci based-HRA approach is a good candidate for broad-band low frequency noise control.

In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations x2±kxy-y2±x=0, where k is a positive integer, and describe the structure of solutions.

The purpose of the paper is to present a new generalization of the dual Fibonacci quaternions called the generalized dual bi-periodic Fibonacci quaternions. This new generalization allow us to state several number of dual quaternion sequences in a unique sequence. Furthermore, we give the generating function, the Binet formula, the norm value and some basic properties of these dual quaternions.

In this paper, a new interconnection network structure called hierarchical extended Fibonacci cubes interconnection networks HEFC1(n) is proposed, and its properties are evaluated. A set of the HEFC1(n)s constructed by the proposed method is a two level conventional hierarchical network. For each n (dimension of extended Fibonacci cube-EFC1(n)), a HEFC1(n)   interconnection network can be constructed. All EFC1(n) for n>4 are recursively constructible, hence HEFC1(n)s for n>4 are also recursively constructible with some extra EFC1(n-1)s and EFC1(n-2)s. Furthermore, since a HEFC1(n) has a hierarchically structured character and the feature of uniformity, a wide variety of inter-cluster connections are possible. The comparisons of HEFC1(n)s with some of the traditional cubic and hierarchical cubic networks are presented in this study. The routing in HEFC1(n+2)s is as easy as routing in HCN(n,n) and the scalability of HEFC1(n+2) is better than the scalability of HCN(n,n) and H(2n). H(2n) and HCN(n,n) are symmetric interconnection networks, however, HEFC1(n+2) is not a symmetric interconnection network and HEFC1(n+2)s have a recurrent structure as H(2n) and HCN(n,n). The cost of HEFC1(n+2) is better than the costs of HCN(n,n) and H(2n).The edge connectivity of HEFC1(n+2) is worse than the edge connectivity of H(2n) and HCN(n,n). This makes the VLSI design of HEFC1(n+2) simpler than the VLSI designs of H(2n) and HCN(n,n).