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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