Repdigits in generalized Pell sequences

Sequences related to the Pell generalized equation

We consider sequences of the type An = 6An−1 − An−2, A0 = r, A1 = s (r and s integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type. Let α = 3 + 2 √ 2, β = 3 − 2 √ 2 be the roots of the polynomial x − 6x+ 1. Note that α+β = 6, αβ = 1, α−β = 4 √ 2. Also let γ = 1+ √ 2...

On The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices

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...

Powers in Recurrence Sequences: Pell Equations

In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the form x −Dy = 1 in integers x, y and n ≥ 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms...

Pell–Padovan-circulant sequences and their applications

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 k-nacci Sequences and The Generalized Order-k Pell Sequences in The Semi-Direct Product of Finite Cyclic Groups