On k-Fibonacci Numbers with Applications to Continued Fractions

2015 International Conference on Mathematics, its Applications, and Mathematics Education

数学;教育

Rabago, Julius Fergy T.^1

Department of Mathematics, Computer Science College of Science, University of the Philippines, Baguio Governor Pack Road, Baguio City

2600, Philippines^1

关键词: Continued fraction;    Continued fraction expansion;    Fibonacci numbers;    Fixed positive integers;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/693/1/012005/pdf
DOI  :  10.1088/1742-6596/693/1/012005

学科分类：发展心理学和教育心理学

来源: IOP

【 摘 要 】

Let (ωn)n∈be the sequence of k-Fibonacci numbers recursively defined by ω1= 1, ω2= 1, ωn+2= kωn+1+ ωn, ∀n ∈, and m be a fixed positive integer. In this work we prove that, for almost every x ∈ (0, 1), the pattern k, k,k (comprising of m-digits) appears in the continued fraction expansion x = [0, a1, a2; ] with frequency f(k,m) := (-1)mk/log 2 log {φm+1-1φm-1}, where φm = ωm+1/ωm, i.e., limn→∞ 1/n #{j ∈ Ωn: aj+i= k for all i ∈ Ωm-1∪{0}} = f(k,m), where Ωn:= {1, 2; , n}.