Linear Recurrent Sequences

2021 ◽  
pp. 29-55
Author(s):  
Masum Billal ◽  
Samin Riasat
Keyword(s):  
Recurrent Sequences ◽  
Linear Recurrent Sequences

scholarly journals On the periodic writing of cubic irrationals and a generalization of Rédei functions

2015 ◽  
Vol 11 (03) ◽  
pp. 779-799 ◽  
Author(s):  
Nadir Murru
Keyword(s):  
Continued Fraction ◽  
Integer Sequences ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences ◽  
Algebraic Properties ◽  
Cubic Polynomial ◽  
Multidimensional Continued Fraction

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

scholarly journals R-Algebras of Linear Recurrent Sequences

1995 ◽  
Vol 175 (1) ◽  
pp. 332-338 ◽  
Author(s):  
U. Cerruti ◽  
F. Vaccarino
Keyword(s):  
Recurrent Sequences ◽  
Linear Recurrent Sequences

On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

2021 ◽  
pp. 1-27
Author(s):  
Harold S. Erazo ◽  
Carlos A. Gómez ◽  
Florian Luca
Keyword(s):  
Pell Equation ◽  
Number Of Solutions ◽  
Pell Equations ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences ◽  
Powers Of 2 ◽  
Integer Solutions

In this paper, we show that if [Formula: see text] is the [Formula: see text]th solution of the Pell equation [Formula: see text] for some non-square [Formula: see text], then given any integer [Formula: see text], the equation [Formula: see text] has at most [Formula: see text] integer solutions [Formula: see text] with [Formula: see text] and [Formula: see text], except for the only pair [Formula: see text]. Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.

scholarly journals On the Lucas property of linear recurrent sequences

2016 ◽  
Vol 13 (06) ◽  
pp. 1617-1625
Author(s):  
Hao Zhong ◽  
Tianxin Cai
Keyword(s):  
Arithmetic Function ◽  
Lucas Numbers ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences ◽  
Fibonacci Sequences

Let [Formula: see text] be an arithmetic function. [Formula: see text] has Lucas property if for any prime [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] In this paper, we discuss the Lucas property of Fibonacci sequences and Lucas numbers. Meanwhile, we find some other interesting results.

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