INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 3, NO. 2, 2013 37 A Note on (k,h)-Jacobsthal Sequence A.C.F. Bueno Abstract—In this paper, we define the (k,h)-Jacobsthal sequence. Furthermore, we derive the formula for the nth term and the sum of the first n terms of this sequence. and Index Terms—Jabobsthal sequence, (k,h)-Jacobsthal sequence, nth term of a sequence, sum of the first n terms of a sequence . This means that MSC 2010 Codes – 11B39, 11B50 HE numbers in the Jacobsthal sequence the recurrence relation given by Jn = Jn−1 + 2Jn−2 ∞ {Jn }n=0 (1) (2) with initial values T0 = 0 and T1 = k where k, h ∈ Z, k and h are not both zero and k 2 + 8h > 0. II. M AIN R ESULTS Theorem 2.1: Tn = k (αn − β n ) p where α= β= k+ k− √ √ (3) k 2 + 8h , 2 k 2 + 8h , 2 k = α + β, and p = α − β. Proof: The recurrence relation of the sequence has the characteristic equation x2 − kx − 2h = 0 whose roots are α= k+ √ k 2 + 8h 2 Tn = c1 αn + c2 β n c1 + c2 = 0 c1 α + c2 β = k satisfy with initial values J0 = 0 and J1 = 1. +∞ In [1], some generalizations on {Jn }n=0 were presented together with some identities. Now we give another generalization of the Jacobsthal sequence and it will be called the +∞ (k,h)-Jacobsthal sequence denoted by {Tn }n=0 . The numbers in this sequence satisfy the recurrence relation Tn = kTn−1 + 2hTn−2 √ Using the initial values T0 = 0 and T1 = k leads to the linear system I. I NTRODUCTION T β= k− k 2 + 8h 2 Aldous Cesar F. Bueno is an instructor in the Department of Mathematics and Physics of the Central Luzon State University, Science City of Mu˜ noz, Nueva Ecija, Phillippines. (e-mail: aldouz [email protected]). k k = kp and c2 = − α−β = − kp . whose solutions are c1 = α−β Hence k (αn − β n ) Tn = p Theorem 2.2: n−1 ∑ m=0 Tm = Tn + 2kTn−1 − k 2h + k − 1 (4) Proof: n−1 ∑ n−1 k ∑ m [α − β m ] p m=0 m=0 [ ] k 1 − αn 1 − βn = − p 1−α 1−β [ ] k (1 − αn )(1 − β) − (1 − β n )(1 − α) = p (1 − α)(1 − β) k = Γ p(2k + k − 1) Tn + 2hTn−1 − k = 2h + k − 1 [ n ] where Γ = (α − β n ) − (α − β) − 2h(αn−1 − β n−1 ) Tm = III. C ONCLUSION In summary, we have obtained the equations for the nth term and the sum of the first n terms of the (k,h)-Jacobsthal sequence. A possible extension of this work is by retaining the recurrence relation while setting the initial values to T0 = a and T = b where a, b ∈ Z and a and b are not both zero. R EFERENCES [1] K. Atanassov, “Short Remarks on Jacobsthal Numbers”, Notes on Number Theory and Discrete Mathematics, vol. 18, no. 2, pp. 63–64, 2012.
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