The aim of this article is to introduce Pell–Padovan generalized quaternions. It also derives new properties associated with these and takes into account negative indices. Additionally, it presents generating function, Binet-like formula, Simson matrix representations, several summation properties.

It is well-known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational α. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r...

The gene pelA encoding a pectate lyase from the strain Bacillus sp. BP-23 was cloned and expressed in Escherichia coli. The nucleotide sequence of a 1214 bp DNA fragment containing pelA gene was determined, revealing an ORF of 666 nucleotides that encoded a protein of 23233 Da. The deduced amino acid sequence of the encoded enzyme showed homology to pectate lyases A, B, C and D from Fusarium so...

Consider the system of Diophantine equations x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to th...

can be reduced to a relation of form (1), between xn and xn+1. Consequently, the relation's two consecutive terms of Fibonacci, Lucas, and Pell sequences are given in [2]. S. Roy [6] finds this relation for the Fibonacci sequence using hyperbolic functions. In this paper we shall prove that if a sequence (xn)n>1 satisfies a linear recurrence of order r > 2, then there exists a polynomial relati...

A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has...

In Part I we met Pell’s equation x2−dy2 = 1 for nonsquare positive integers d. We stated Lagrange’s theorem that every Pell equation has a nontrivial solution (an integral solution besides (±1, 0)) and saw what all solutions to Pell’s equation look like if there’s a nontrivial solution. As in Part I, “solution” means integral solution. Here we will prove Lagrange’s theorem in Section 2 and show...

In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, k-Jacobsthal and p-numbers, distance numbers, (2, k)-distance generalized (k, r)-Fibonacci in the sense by extending definition a recurrence relation with two adding three distance, simultaneously. Tiling combinatorial interp...