scholarly journals Are the Catalan Numbers a Linear Recurrence Sequence?

Martin Klazar ◽  
Richard Horský

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.

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Lan Qi ◽  
Zhuoyu Chen

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.

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 788 ◽  
Zhuoyu Chen ◽  
Lan Qi

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.

2016 ◽  
pp. 1-30
Alexi Block Gorman ◽  
Tyler Genao ◽  
Heesu Hwang ◽  
Noam Kantor ◽  
Sarah Parsons ◽  

2012 ◽  
Vol 15 ◽  
pp. 360-373
Jonas Šiurys

AbstractWe prove that for each positive integerkin the range 2≤k≤10 and for each positive integerk≡79 (mod 120) there is ak-step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2152
Marie Hubálovská ◽  
Štěpán Hubálovský ◽  
Eva Trojovská

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.

Japhet Odjoumani ◽  
Volker Ziegler

AbstractIn this paper we consider the Diophantine equation $$U_n=p^x$$ U n = p x where $$U_n$$ U n is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on $$U_n$$ U n , we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.

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