In this paper, we give some relations involving the usual Fibonacci and generalized order-k Pell numbers. These relations show that the generalized order-k Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We …nd families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, F2i 1; and their sums. Also extending these mat...

In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence...

The main object of the study is to consider binomial transform for quadra Fibona-Pell sequence and quaternion. In paper, which consists two parts in terms results found, first step was taken by defining part then finding recurrence relation this new transform. Then, Binet formula, generating function various sum formulas were found. second part, applied quaternion, discussed a thesis before. Si...

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, thenwe define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix me...

Abstract In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known sequences, as Fibonacci sequence, Lucas Pell Pell-Lucas etc. We limit formulas, where product first n terms these appears. Furthermore, prove limits connect to number e (? 2:71828:::).

In this paper, we present a generalization of well-known k-Fibonacci sequence. Namely, defined generalized This sequence generalizes others, sequence, classical Fibonacci Pell and Jacobsthal We establish some the interesting properties Also, obtain generating function for them.

(iv) un = (a b)/(a b), yn = a + £. If i4 = £ = 1, then un, Vn are the Fibonacci and Lucas sequences, respectively. If A = 3 and 5 =-2, then un, t;n are the Mersenne and Fermat sequences, respectively. If A = 2 and 5 = 1 (so that Z? = 8), then un is called the Pell sequence (see [4, p. 187]), and is denoted Pn; vn may be called the secondary Pell sequence, and denoted i?n, following [7]. For the...

In 2009, Luca and Nicolae [[Formula: see text], Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another number are [Formula: text]. 2015, Faye [Pell a Pell number, Publ. Inst. Math. 101(115) (2017) 231–245] text] Here, we add to these two results prove for any fixed natural if define sequence as all then solution Diophantine equation