In this study, we consider the quadratic Diophantine equations given in the title and determine when these equations have positive integer solutions. Moreover, we find all positive integer solutions of them in terms of Balancing numbers Bn, Pell and Pell-Lucas numbers, and the terms of the sequence {vn} , where {vn} is defined by v0 = 2, v1 = 6, and vn+1 = 6vn − vn−1 for n ≥ 1.

The aim of this paper is to introduce a generalization Modified Pell numbers. Some identities about new sequence are obtained and also investigated some relationships with another sequence. Finally, using these sequences the row column norms Hankel matrices presented.

In this paper we present new families of sequences that generalize the Jacobsthal and the Jacobsthal-Lucas numbers and establish some identities. We also give a generating function for a particular case of the sequences presented. Introduction Several sequences of positive integers were and still are object of study for many researchers. Examples of these sequences are the well known Fibonacci ...

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...

We describe how to compute the intersection of two Lucas sequences of the forms {Un(P,±1)}n=0 or {Vn(P,±1)}n=0 with P ∈ Z that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case Un(1,−1) and Un(3, 1) and the case of two V -sequences when the product of their discriminants is a perfect square. Moreover, the inter...

Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and d = k − k. In the first section we give some preliminaries from Pell equations x − dy = 1 and x − dy = N , where N be any fixed positive integer. In the second section, we consider the integer solutions of Pell equations x − dy = 1 and x − dy = 2. We give a method for the solutions of these equations. Further we derive recurrence relations...

In this paper we generalize the Pell numbers and the Pell-Lucas numbers and next we give their graph representations. We shall show that the generalized Pell numbers and the generalized Pell-Lucas numbers are equal to the total number of independent sets in special graphs.