The bi-periodic Fibonacci sequence also known as the generalized Fibonacci sequence was fırst introduced into literature in 2009 by Edson and Yayenie [9] after which the bi-periodic Lucas sequence was defined in a similar fashion in 2004 by Bilgici [5]. In this study, we introduce a new generalization of the Jacobsthal numbers which we shall call bi-periodic Jacobsthal sequences similar to the ...

The infinite Fibonacci sequence F, which is an extension of the classic Fibonacci sequence to the infinite alphabet N, is the fixed point of the morphism φ: (2i) 7→ (2i)(2i+ 1) and (2i+ 1) 7→ (2i+ 2) for all i ∈ N. In this paper, we study the growth order and digit sum of F, and give several decompositions of F using singular words.

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence . It is shown that this sequence is periodic with the period ( ) if | | .

For a finitely generated group G = 〈A〉, where A = {a1, a2, . . . , an}, the sequence xi = ai, 1 ≤ i ≤ n, xi+n = ∏n j=1 xi+j−1, i ≥ 1, is called the Fibonacci orbit of G with respect to the generating set A, denoted FA(G). If FA(G) is periodic we call the length of the period of the sequence the Fibonacci length of G with respect to A, written LENA(G). In this paper we examine the Fibonacci leng...

In this paper, we give formulas for the sums of generalized order-k Fibonacci, Pell and similar other sequences which we obtain using matrix methods. As applications, we give explicit formulas for the Tribonacci and Tetranacci numbers. 1. Introduction The well-known Fibonacci sequence fFng is de…ned for n > 2 by Fn+1 = Fn + Fn 1 where F1 = F2 = 1: The well-known Pell sequence fPng is de…ned for...

The k-generalized Fibonacci sequence ( F (k) n ) n≥2−k is the linear recurrent sequence of order k, whose first k terms are 0, . . . , 0, 1 and each term afterwards is the sum of the preceding k terms. Two or more terms of a k-generalized Fibonacci sequence are said to be in the same power of two-class if the largest odd factors of the terms are identical. In this paper, we show that for each k...