Notes on Number Theory and Discrete Mathematics ISSN 1310–5132 Vol. 21, 2015, No. 1, 67–69
A note on generalized Tribonacci sequence Aldous Cesar F. Bueno Department of Mathematics and Physics, Central Luzon State University Science City of Mu˜noz 3120, Nueva Ecija, Philippines e-mail:
[email protected]
Abstract: In this study, we provide a property of the generalized Tribonacci sequence through limits. Keywords: Tribonacci sequence, Generalized Tribonacci sequence, Tribonacci constant. AMS Classification: 11B39.
1
Introduction
The Tribonacci sequence {Tk }+∞ k=0 satisfy the recurrence relation Tn = Tn−1 + Tn−2 + Tn−3 , having the initial values T0 = 0, T1 = 0 and T2 = 1. Furthermore, its Binet’s formula is given by Tn =
σ n+1 ρn+1 τ n+1 + + , (τ − σ)(τ − ρ) (σ − τ )(σ − ρ) (ρ − τ )(ρ − σ)
where τ , σ and ρ are the roots of x3 − x2 − x − 1 = 0. These roots have the following properties: σn = 0, n→+∞ τ n lim
ρn = 0, n→+∞ τ n lim
and due to these we have,
Tn+1 = τ. n→+∞ Tn lim
Actually, the root τ is called the Tribonacci constant and its explicit form is p p √ √ 3 3 1 + 19 + 3 33 + 19 − 3 33 τ= . 3 67
On the other hand, the generalized Tribonacci sequence denoted by {Sk }+∞ k=0 satisfy the recurrence relation Sn = Sn−1 + Sn−2 + Sn−3 , where the initial values S1 , S2 and S3 are arbitrary but are not simultaneously zero. Natividad and Policarpio [1] provided a formula for finding its n-th term and it is given by Sn = Tn−2 S1 + (Tn−2 + Tn−3 )S2 + Tn−1 S3 . Our goal in this study is to investigate the generalized Tribonacci sequence through limits specifically, where we will be dealing on the limit given by Sn+j , n→+∞ Sn lim
where j is a positive integer.
2
Main results
Theorem.
Sn+j = τ j. n→+∞ Sn lim
Proof: Sn+j n→+∞ Sn Tn+j−2 S1 + (Tn+j−2 + Tn+j−3 )S2 + Tn+j−1 S3 = lim n→+∞ Tn−2 S1 + (Tn−2 + Tn−3 )S2 + Tn−1 S3 Tn+j−2 S1 /Tn + (Tn+j−2 + Tn+j−3 )S2 /Tn + Tn+j−1 S3 /Tn = lim . n→+∞ Tn−2 S1 /T − n + (Tn−2 + Tn−3 )S2 /Tn + Tn−1 S3 /Tn lim
Claim: lim Tn+j /Tn = τ j
n→+∞
lim Tn+j /Tn
n→+∞
= lim
n→+∞
= =
τ n+j+1 (τ −σ)(τ −ρ)
+
σ n+j+1 (σ−τ )(σ−ρ)
+
τ n+1 (τ −σ)(τ −ρ)
+
σ n+1 (σ−τ )(σ−ρ)
+
τ n+j+1 /τ n (τ −σ)(τ −ρ) lim n+1 /τ n n→+∞ τ (τ −σ)(τ −ρ) j+1
τ
τ j =τ .
+ +
σ n+j+1 /τ n (σ−τ )(σ−ρ) σ n+1 /τ n (σ−τ )(σ−ρ)
+ +
ρn+j+1 (ρ−τ )(ρ−σ) ρn+1 (ρ−τ )(ρ−σ) ρn+j+1 /τ n (ρ−τ )(ρ−σ) ρn+1 /τ n (ρ−τ )(ρ−σ)
; by the properties of the roots τ, σ, and ρ
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Continuing and using our proven claim, we obtain τ j−2 S1 + (τ j−2 + τ j−3 )S2 + τ j−1 S3 τ −2 S1 + (τ −2 + τ −3 )S2 + τ −1 S3 τ j+1 S1 + (τ j+1 + τ j )S2 + τ j+2 S3 = τ S1 + (τ + 1)S2 + τ 2 S3 τ j [τ S1 + (τ + 1)S2 + τ 2 S3 ] = τ S1 + (τ + 1)S2 + τ 2 S3 = τ j,
as desired. Corollary. Sn+1 = τ. n→+∞ Sn lim
3
Conclusion
In summary, we have established that like the Tribonacci sequence, the ratio of the two terms S Sn+j and Sn of the generalized Tribonacci sequence, given by Sn+j , approaches the value τ j as n n tends to infinity where τ is the Tribonacci constant.
References [1] Natividad, L. R. & Policarpio, P. B. (2013) A novel formula in solving Tribonacci-like sequence, General Mathematics Notes, 17(1), 82–87. [2] Piezas III, T. (2010) A tale of four constants, https://sites.google.com/site/ tpiezas/0012
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