A new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power Pnl is expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R=2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.
In this paper we introduce a new kind of generalized Jacobsthal numbers in a distance sense. We give the identities and matrix representations for them and their connections with the Fibonacci and the Pell numbers. We also describe the interpretations of these numbers in terms of some kind of (k1A1, k2A2, k3A3)-edge colouring and quasi colouring.
In this paper, we define and study a new one-parameter generalization of the Pell hybrid numbers. Based on the definition of r-Pell numbers, we define the r-Pell hybrid numbers. We give their properties: character, Binet formula, summation formula, and generating function. Moreover, we present Catalan, Cassini, d’Ocagne, and Vajda type identities for the r-Pell hybrid numbers.
In this paper we present a new one parameter generalization of the classical Pell numbers. We investigate the generalized Binet’s formula, the generating function and some identities for r-Pell numbers. Moreover, we give a graph interpretation of these numbers.
In this paper, we obtain a closed form for FΣ_{i=1}^k, PΣ_{i=1}^k and JΣ_{i=1}^k for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases FΣ_{i=1}^n, PΣ_{i=1}^n and JΣ_{i=1}^n for any arbitrary positive integer n.
2019-06-13
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English