A note on the sums of reciprocal k-Fibonacci numbers of subscript

In this article we find the fïnite sum of reciprocal k-Fibonacci numbers of subscript 2n a, then we fïnd the infinite sum of these numbers. Special cases of these sums  for the classical Fibonacci sequence and the Pell sequence are indicated. Finally we propose a new way to fïnd the infinite sum of the reciprocal k-Fibonacci numbers with odd subscripts and, consequently, the sum of all reciprocal k-Fibonacci numbers, but without finding the answer to this problem (Erdos)

k-Balancing Numbers and Pell’s Equation of Higher Orde

First time introduced in the year 1999, the balancing numbers are extensively studied. Each balancing number is associated with a Lucas-balancing number and are useful in the computation of balancing numbers of higher order. In this report, we study the sums of k-balancing numbers with indexes in an arithmetic sequence, say an+r for fixed integers a and r. Also an infinite family of Pell’s equations of degree n>2 are discussed

Combinatorial structure of cube of fuzzy topographic topological mapping and K-fibonacci sequence

Fuzzy Topographic Topological Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four topological spaces that are homeomorphic to each other. A sequence of FTTMn is a combination of n terms of FTTM. In previous studies, FTTM are linked with three mathematical concepts namely; FTTM with Pascal’s Triangle, FTTM as a graph and FTTM in relation to k-Fibonacci sequence. In this research, the relationship between graph of FTTMn and k-Fibonacci is established via Hamiltonian polygonal paths in an assembly graph of FTTMn. The assembly graph is a graph with all vertices have valency of one or four. The Hamiltonian path is a path that visits every vertex of a graph exactly once. The structure of assembly graph of FTTMn including maximal assembly graph of FTTMn is introduced and its properties are investigated. The existence of Hamiltonian polygonal path in maximal assembly graph of FTTMn is proven. Several new definitions and theorems for the assembly graph of FTTMn and Hamiltonian polygonal path in maximal assembly graph of FTTMn are stated and proven, respectively. Finally, a theorem that highlight the relation between graph of FTTMn to k-Fibonacci sequence is proven

On the Spectral Norms of r -Circulant Matrices with the k -Fibonacci and k -Lucas Numbers

Abstract In this paper, we consider the k -Fibonacci and k -Lucas sequences be r -circulant matrices. Afterwards, we give upper and lower bounds for the spectral norms of matrices A and B. In addition, we obtain some bounds for the spectral norms of Hadamard and Kronecker products of these matrices. Mathematics Subject Classification: 15A45, 15A6