Pellans sequence and its diophantine triples

We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it ?pellans sequence? and then we solve the system ab+1=Sx, ac+1=Sy bc+1=Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

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Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽

10.3390/sym11121476 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1476 ◽

Cited By ~ 1

Author(s):

Lan Qi ◽

Zhuoyu Chen

Keyword(s):

Power Series ◽

Series Expansion ◽

Generating Function ◽

Fourth Order ◽

Power Series Expansion ◽

Linear Recurrence ◽

Recurrence Sequence ◽

Order Linear ◽

Elementary Methods ◽

Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

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Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Symmetry ◽

10.3390/sym11060788 ◽

2019 ◽

Vol 11 (6) ◽

pp. 788 ◽

Cited By ~ 7

Author(s):

Zhuoyu Chen ◽

Lan Qi

Keyword(s):

Power Series Expansion ◽

Second Order ◽

Linear Recurrence ◽

Recurrence Sequence ◽

Order Linear ◽

Symmetry Properties ◽

Elementary Methods ◽

Linear Recurrence Sequence ◽

Convolution Formula ◽

The Relationship

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

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On Homogeneous Combinations of Linear Recurrence Sequences

Mathematics ◽

10.3390/math8122152 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2152

Author(s):

Marie Hubálovská ◽

Štěpán Hubálovský ◽

Eva Trojovská

Keyword(s):

Fibonacci Sequence ◽

Linear Recurrence ◽

Degree Polynomial ◽

Recurrence Sequence ◽

Linear Recurrence Sequence ◽

Recurrence Sequences ◽

Positive Integers

Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2=Fn+1+Fn, for n≥0, where F0=0 and F1=1. There are several interesting identities involving this sequence such as Fn2+Fn+12=F2n+1, for all n≥0. In 2012, Chaves, Marques and Togbé proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and Gn+1s+⋯+Gn+ℓs∈(Gm)m, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of (Gm)m. In this paper, we shall prove that if P(x1,…,xℓ) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(Gn+1,…,Gn+ℓ)∈(Gm)m for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of (Gm)m and the coefficients of P.

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ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720001094 ◽

2020 ◽

pp. 1-10

Author(s):

CLEMENS FUCHS ◽

SEBASTIAN HEINTZE

Keyword(s):

Number Field ◽

Function Field ◽

Function Fields ◽

Linear Recurrence ◽

Recurrence Sequence ◽

Linear Recurrences ◽

Field Case ◽

Power Sum ◽

Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

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NONLINEAR FILTRATION OF BINARY LINEAR RECURRENCE SEQUENCE WITH RANDOM DELAY AND RANDOM INITIAL PHASE

Telecommunications and Radio Engineering ◽

10.1615/telecomradeng.v78.i6.20 ◽

2019 ◽

Vol 78 (6) ◽

pp. 475-487

Author(s):

M. G. Bakulin ◽

V. B. Kreyndelin ◽

D. Yu. Pankratov

Keyword(s):

Initial Phase ◽

Linear Recurrence ◽

Nonlinear Filtration ◽

Random Delay ◽

Recurrence Sequence ◽

Linear Recurrence Sequence

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On the fourth-order linear recurrence formula related to classical Gauss sums

Open Mathematics ◽

10.1515/math-2017-0104 ◽

2017 ◽

Vol 15 (1) ◽

pp. 1251-1255 ◽

Cited By ~ 14

Author(s):

Chen Zhuoyu ◽

Zhang Wenpeng

Keyword(s):

Positive Integer ◽

Fourth Order ◽

Recurrence Formula ◽

Character Sums ◽

Gauss Sums ◽

Linear Recurrence ◽

Analytic Method ◽

Order Linear ◽

Computational Problem

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.

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