In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra $\mathbb{O}(a,b,c)$ where $a,b$ $c$ are real numbers. For these number sequences, obtain Binet formulas some well-known identities such as Catalan's identity, Cassini's identity d'Ocagne's identity.

Five endopectate lyases from the phytopathogenic bacterium Erwinia chrysanthemi, PelA, PelB, PelD, PelI, and PelL, were analyzed with respect to their modes of action on polymeric and oligomeric substrates (degree of polymerization, 2 to 8). On polygalacturonate, PelB showed higher reaction rates than PelD, PelI, and PelA, whereas the reaction rates for PelL were extremely low. The product prog...

This paper develops properties of recurrence sequences defined from circulant matrices obtained from the characteristic polynomial of the Pell–Padovan sequence. The study of these sequences modulo m yields cyclic groups and semigroups from the generating matrices. Finally, we obtain the lengths of the periods of the extended sequences in the extended triangle groups E(2, n, 2), E(2, 2, n) and E...

The purpose of this paper is twofold. As the first goal, we show that three different classes of random walks are counted by the Pell numbers. The calculations are done using a convenient technique that involves the Riordan group. This leads to the second goal, which is to demonstrate this convenient technique. We also construct bijections among Pell, certain Motzkin, and certain Schröder walks...

The main aim of this work is to introduce the Gaussian Pell quaternion QGpn and Pell-Lucas QGqn, where components QGqn are numbers pn qn, respectively. Firstly, we obtain recurrence relations Binet formulas for QGqn. We use prove Cassini?s identity these quaternions. Furthermore, give some basic identities such as summation formulas, terms with negative indices generating functions complex

In this study, we define unrestricted Pell and – Lucas hyper-complex numbers. We choose arbitrary numbers for the coefficients of ordered basis 〖{e〗_0,e_1,⋯,e_(N-1)} 2^N-ons where N∈{0,1,2,3,4} call these Pell-Lucas 2N-ons. give generating functions Binet formulas type also obtain some generalization well known identities such as Catalan’s, Cassini’s d’Ocagne’s identities.

In a recent note, Santana and Diaz–Barrero proved a number of sum identities involving the well–known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the the electronic journal of combinatorics 13 (2006), #R00 1 results by in...