New families of Jacobsthal and Jacobsthal-Lucas numbers
In this paper we present new families of sequences that generalize the Jacobsthal and the Jacobsthal-Lucas numbers and establish some identities. We also give a generating function for a particular case of the sequences presented.
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irk-123456789-1547562019-06-16T01:31:24Z New families of Jacobsthal and Jacobsthal-Lucas numbers Catarino, P. Vasco, P. Campos, H. Aires, A.P. Borges, A. In this paper we present new families of sequences that generalize the Jacobsthal and the Jacobsthal-Lucas numbers and establish some identities. We also give a generating function for a particular case of the sequences presented. 2015 Article New families of Jacobsthal and Jacobsthal-Lucas numbers / P. Catarino, P. Vasco, H. Campos, A.P. Aires, A. Borges // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 40-54 . — Бібліогр.: 21 назв. — англ. 1726-3255 2010 MSC:11B37, 11B83, 05A15. http://dspace.nbuv.gov.ua/handle/123456789/154756 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we present new families of sequences that generalize the Jacobsthal and the Jacobsthal-Lucas numbers and establish some identities. We also give a generating function for a particular case of the sequences presented. |
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Article |
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Catarino, P. Vasco, P. Campos, H. Aires, A.P. Borges, A. |
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Catarino, P. Vasco, P. Campos, H. Aires, A.P. Borges, A. New families of Jacobsthal and Jacobsthal-Lucas numbers Algebra and Discrete Mathematics |
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Catarino, P. Vasco, P. Campos, H. Aires, A.P. Borges, A. |
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Catarino, P. |
| title |
New families of Jacobsthal and Jacobsthal-Lucas numbers |
| title_short |
New families of Jacobsthal and Jacobsthal-Lucas numbers |
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New families of Jacobsthal and Jacobsthal-Lucas numbers |
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New families of Jacobsthal and Jacobsthal-Lucas numbers |
| title_full_unstemmed |
New families of Jacobsthal and Jacobsthal-Lucas numbers |
| title_sort |
new families of jacobsthal and jacobsthal-lucas numbers |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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New families of Jacobsthal and Jacobsthal-Lucas numbers / P. Catarino, P. Vasco, H. Campos, A.P. Aires, A. Borges // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 40-54 . — Бібліогр.: 21 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT catarinop newfamiliesofjacobsthalandjacobsthallucasnumbers AT vascop newfamiliesofjacobsthalandjacobsthallucasnumbers AT camposh newfamiliesofjacobsthalandjacobsthallucasnumbers AT airesap newfamiliesofjacobsthalandjacobsthallucasnumbers AT borgesa newfamiliesofjacobsthalandjacobsthallucasnumbers |
| first_indexed |
2025-07-14T06:51:36Z |
| last_indexed |
2025-07-14T06:51:36Z |
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1837604165764775936 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 1, pp. 40–54
© Journal “Algebra and Discrete Mathematics”
New families of Jacobsthal and
Jacobsthal-Lucas numbers
Paula Catarino, Paulo Vasco, Helena Campos,
Ana Paula Aires and Anabela Borges
Communicated by V. Mazorchuk
Abstract. In this paper we present new families of sequences
that generalize the Jacobsthal and the Jacobsthal-Lucas numbers
and establish some identities. We also give a generating function
for a particular case of the sequences presented.
Introduction
Several sequences of positive integers were and still are object of study
for many researchers. Examples of these sequences are the well known
Fibonacci sequence and the Lucas sequence, both related with the golden
mean, with so many applications in diverse fields such as mathematics,
engineering, biology, physics, architecture, stock market investing, among
others (see [9] and [17]). About these and other sequences like Pell sequence,
Pell-Lucas sequence, Modified Pell sequence, Jacobsthal sequence and the
Jacobsthal-Lucas sequence, among others, there is a vast literature where
several properties are studied and well known identities are derived, see
for example, [13, 18–20].
In 1965, Horadam studied some properties of sequences of the type,
wn(a, b; p, q), where a, b are nonnegative integers and p, q are arbitrary
2010 MSC: 11B37, 11B83, 05A15.
Key words and phrases: Jacobsthal numbers, Jacobsthal-Lucas numbers, Binet
formula, Generating matrix, Generating function.
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 41
integers, see [11] and [12]. Such sequences are defined by the recurrence
relations of second order
wn = pwn−1 − qwn−2, (n > 2)
with initial conditions w0 = a, w1 = b. For example, the Fibonacci and
the Lucas sequences can be considered as special cases of sequences of this
type, wn(1, 1; 1, −1) and wn(2, 1; 1, −1), respectively. Also, the Jacobsthal
and the Jacobsthal-Lucas sequences can be considered as wn(0, 1; 1, −2)
and wn(2, 1; 1, −2), respectively. Recall that the second-order recurrence
relations and the initial conditions for the Jacobsthal numbers, Jn, n > 0,
and for the Jacobsthal-Lucas numbers, jn, n > 0, respectively, are given
by
Jn+2 = Jn+1 + 2Jn, J0 = 0, J1 = 1
and
jn+2 = jn+1 + 2jn, j0 = 2, j1 = 1.
The Binet formulae for these sequences are
Jn =
2n − (−1)n
3
and jn = 2n + (−1)n,
where 2 and −1 are the roots of the characteristic equation associated
with the above recurrence relations.
More recently, some of these sequences were generalized for any pos-
itive real number k. The studies of k-Fibonacci sequence, k-Lucas se-
quence, k-Pell sequence, k-Pell-Lucas sequence, Modified k-Pell sequence,
k-Jacobhstal and k-Jacobhstal-Lucas sequence, can be found in [1,3–7,14].
The aim of this work is to study some properties of two new sequences
that generalize the Jacobhstal and the Jacobsthal-Lucas numbers. In
this work we will follow closely the work of El-Mikkawy and Sogabe (see
[10]) where the authors give a new family that generalizes the Fibonacci
numbers, different from the one defined in [1], and establish relations with
the ordinary Fibonacci numbers.
So, in this Section we start giving the new definition of generalized
Jacobsthal and Jacobsthal-Lucas numbers, and we exhibit some elements
of them. We also present relations of these sequences with ordinary
Jacobsthal and Jacobsthal-Lucas. In Section 1 we deduce some properties
of these new families, as well as in Section 2, but using different methods.
In Section 3, we study a particular case, that is two sequences of the new
defined families for k = 2. For these sequences we present some recurrence
relations and generating functions.
42 New families of Jacobsthal and Jacobsthal-Lucas numbers
Following our ideas, we give a new definition of generalized Jacobsthal
and Jacobsthal-Lucas numbers.
Definition 1. Let n be a nonnegative integer and k be a natural number.
By the division algorithm there exist unique numbers m and r such that
n = mk + r (0 6 r < k). Using these parameters we define the new
generalized Jacobsthal and generalized Jacobsthal-Lucas numbers, J
(k)
n
and j
(k)
n respectively by
J (k)
n =
1
(r1 − r2)k
(
rm+1
1 − rm+1
2
)r
(rm
1 − rm
2 )k−r (1)
and
j(k)
n =
(
rm+1
1 + rm+1
2
)r
(rm
1 + rm
2 )k−r
, (2)
where r1 = 2, r2 = −1, respectively.
For k = 1, 2, 3 the first seven elements of these new sequences are:
{
J
(1)
n
}5
n=0
= {0, 1, 1, 3, 5, 11, 21}
{
j
(1)
n
}5
n=0
= {2, 1, 5, 7, 17, 31, 65}
{
J
(2)
n
}5
n=0
= {0, 0, 1, 1, 1, 3, 9}
{
j
(2)
n
}5
n=0
= {4, 2, 1, 5, 25, 35, 49}
{
J
(3)
n
}5
n=0
= {0, 0, 0, 1, 1, 1, 1}
{
j
(3)
n
}5
n=0
= {8, 4, 2, 1, 5, 25, 125} .
We also present more elements of some of these new sequences in the
tables 1 and 2. We have found some interesting regularities. In the case
of the generalized Jacobsthal sequences
{
J
(k)
n
}
n
it is easy to prove that:
Proposition 1. Let J
(k)
i be the ith term of the new family of Jacobsthal
numbers. Then we have:
a) J
(k)
i = 0, i ∈ {0, . . . , k − 1} ;
b) J
(k)
i = 1, i ∈ {k, . . . , k − 1} ;
c) J
(k)
i = 3i−2k, i ∈ {2k, . . . , 3k} .
To the generalized Jacobsthal-Lucas sequences
{
j
(k)
n
}
n
is easy to prove
that:
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 43
Table 1. J
(k)
n , for k = 1, 2, . . . , 9 and n = 0, 1, . . . , 27.
n\k 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0
2 1 1 0 0 0 0 0 0 0
3 3 1 1 0 0 0 0 0 0
4 5 1 1 1 0 0 0 0 0
5 11 3 1 1 1 0 0 0 0
6 21 9 1 1 1 1 0 0 0
7 43 15 3 1 1 1 1 0 0
8 85 25 9 1 1 1 1 1 0
9 171 55 27 3 1 1 1 1 1
10 341 121 45 9 1 1 1 1 1
11 683 231 75 27 3 1 1 1 1
12 1365 441 125 81 9 1 1 1 1
13 2731 903 275 135 27 3 1 1 1
14 5461 1849 605 225 81 9 1 1 1
15 10923 3655 1331 375 243 27 3 1 1
16 21845 7225 2541 625 405 81 9 1 1
17 43691 14535 4851 1375 675 243 27 3 1
18 87381 29241 9261 3025 1125 729 81 9 1
19 174763 58311 18963 6655 1875 1215 243 27 3
20 349525 116281 38829 14641 3125 2025 729 81 9
21 699051 232903 79507 27951 6875 3375 2187 243 27
22 1398101 466489 157165 53361 15125 5625 3645 729 81
23 2796203 932295 310675 101871 33275 9375 6075 2187 243
24 5592405 1863225 614125 194481 73205 15625 10125 6561 729
25 11184811 3727815 1235475 398223 161051 34375 16875 10935 2187
26 22369621 7458361 2485485 815409 307461 75625 28125 18225 6561
27 44739243 14913991 5000211 1669647 586971 166375 46875 30375 19683
Table 2. j
(k)
n , for k = 1, 2, . . . , 9 and n = 0, 1, . . . , 27.
n\k 1 2 3 4 5 6 7 8 9
0 2 4 8 16 32 64 128 256 512
1 1 2 4 8 16 32 64 128 256
2 5 1 2 4 8 16 32 64 128
3 7 5 1 2 4 8 16 32 64
4 17 25 5 1 2 4 8 16 32
5 31 35 25 5 1 2 4 8 16
6 65 49 125 25 5 1 2 4 8
7 127 119 175 125 25 5 1 2 4
8 257 289 245 625 125 25 5 1 2
9 511 527 343 875 625 125 25 5 1
10 1025 961 833 1225 3125 625 125 25 5
11 2047 2015 2023 1715 4375 3125 625 125 25
12 4097 4225 4913 2401 6125 15625 3125 625 125
13 8191 8255 8959 5831 8575 21875 15625 3125 625
14 16385 16129 16337 14161 12005 30625 78125 15625 3125
15 32767 32639 29791 34391 16807 42875 109375 78125 15625
16 65537 66049 62465 83521 40817 60025 153125 39062 78125
17 131071 131327 130975 152303 99127 84035 214375 546875 390625
18 262145 261121 274625 277729 240737 117649 300125 765625 1953125
19 524287 523775 536575 506447 584647 285719 420175 1071875 2734375
20 1048577 1050625 1048385 923521 1419857 693889 588245 1500625 3828125
21 2097151 2098175 2048383 1936415 2589151 1685159 823543 2100875 5359375
22 4194305 4190209 4145153 4060225 4721393 4092529 2000033 2941225 7503125
23 8388607 8386559 8388223 8513375 8609599 9938999 4857223 4117715 10504375
24 16777217 16785409 16974593 17850625 15699857 24137569 11796113 5764801 14706125
25 33554431 33558527 33751039 34877375 28629151 44015567 28647703 14000231 20588575
26 67108865 67092481 67108097 68145025 60028865 80263681 69572993 34000561 28824005
27 134217727 134209535 133432831 133144895 125866975 146363183 168962983 82572791 40353607
44 New families of Jacobsthal and Jacobsthal-Lucas numbers
Proposition 2. Let j
(k)
i be the ith term of the new family of Jacobsthal-
Lucas numbers. Then we have:
a) j
(k)
i = 2k−i, i ∈ {0, . . . , k − 1} ;
b) j
(k)
i = 5i−k, i ∈ {k, . . . , 2k} .
The generalized Jacobsthal and Jacobsthal-Lucas numbers have the
following relations with the ordinary Jacobsthal and Jacobsthal-Lucas
numbers.
Lemma 1. Given n a nonnegative integer and k a natural number
J
(k)
mk+r = (Jm)k−r(Jm+1)r
and
j
(k)
mk+r = (jm)k−r(jm+1)r,
where m and r are nonnegative integers such that n = mk +r (0 6 r < k).
Proof. We have
(Jm)k−r(Jm+1)r =
(
2m − (−1)m
3
)k−r
(
2m+1 − (−1)m+1
3
)r
=
1
3k
(2m − (−1)m)k−r
(
2m+1 − (−1)m+1
)r
=
1
(r1 − r2)k
(rm
1 − rm
2 )k−r
(
rm+1
1 − rm+1
2
)r
= J
(k)
mk+r.
In a similar way we can show the second equality.
Note that the use of the Lemma 1 allows us to conclude immediately
that J
(1)
n and j
(1)
n are the Jacobsthal and the Jacobsthal-Lucas numbers,
respectively.
1. Properties
Next we present some properties of these new families of integers.
Theorem 1. Let k and m be fixed numbers where m is a nonnegative
integer and k a natural number. The generalized Jacobsthal numbers and
the ordinary Jacobsthal numbers satisfy:
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 45
a)
k−1
∑
i=0
(k−1
i
)
(−1)−iJ
(k)
mk+i = (−2)k−1JmJ
(k−1)
(m−1)(k−1);
b)
k−1
∑
i=0
(k−1
i
)
2k−i−1J
(k)
mk+i = JmJ
(k−1)
(m+2)(k−1);
c)
k−1
∑
i=0
J
(k)
mk+i = Jm
2Jm−1
(
J
(k)
(m+1)k − J
(k)
mk
)
.
Proof. a) By Lemma 1 we have that
k−1
∑
i=0
(k−1
i
)
(−1)−iJ
(k)
mk+i is successively
equal to
k−1
∑
i=0
(
k − 1
i
)
(−1)−i(−1)k−1(−1)1−k(Jm)k−i(Jm+1)i
= (−1)1−k
k−1
∑
i=0
(
k − 1
i
)
(−1)k−1−i(Jm)k−1−iJm(Jm+1)i
= (−1)1−kJm
k−1
∑
i=0
(
k − 1
i
)
(−Jm)k−1−i(Jm+1)i,
that by the binomial theorem is equal to
(−1)1−kJm (Jm+1 − Jm)k−1
.
Since, by the definition of the Jacobsthal sequence,
(−1)1−kJm (Jm+1 − Jm)k−1 = (−1)1−kJm (2Jm−1)k−1
and using Lemma 1 (considering m − 1 instead of m, k − 1 instead of k
and r = 0) we obtain
(−1)1−k2k−1JmJ
(k−1)
(m−1)(k−1),
and the result follows.
b) By Lemma 1 we have
k−1
∑
i=0
(
k − 1
i
)
2k−i−1J
(k)
mk+i =
k−1
∑
i=0
(
k − 1
i
)
(Jm)k−i2k−i−1(Jm+1)i
=
k−1
∑
i=0
(
k − 1
i
)
(2Jm)k−i−1Jm(Jm+1)i
= Jm
k−1
∑
i=0
(
k − 1
i
)
(2Jm)k−i−1(Jm+1)i
46 New families of Jacobsthal and Jacobsthal-Lucas numbers
and using again the binomial theorem we have
Jm(Jm+1 + 2Jm)k−1,
that is equal, by the definition of the Jacobsthal numbers, to
Jm(Jm+2)k−1
and by Lemma 1 (considering m + 2 instead of m, k − 1 instead of k and
r = 0), we get
JmJ
(k−1)
(m+2)(k−1).
c) By Lemma 1 we can write
k−1
∑
i=0
J
(k)
mk+i =
k−1
∑
i=0
(Jm)k−i(Jm+1)i
= (Jm)k
k−1
∑
i=0
(
Jm+1
Jm
)i
= (Jm)k
(
Jm+1
Jm
)k
− 1
Jm+1
Jm
− 1
= (Jm)k
[
(Jm+1)k − (Jm)k
(Jm)k
×
Jm
Jm+1 − Jm
]
=
Jm
Jm+1 − Jm
[
(Jm+1)k − (Jm)k
]
=
Jm
2Jm−1
[
(Jm+1)k − (Jm)k
]
and, taking into account Lemma 1 (with r = 0), the result follows.
The following result for Jacobsthal-Lucas numbers can be deduced
analogously:
Theorem 2. Let k and m be fixed numbers where m is a nonnegative
integer and k a natural number. The generalized Jacobsthal-Lucas numbers
and the ordinary Jacobsthal-Lucas numbers satisfy:
a)
k−1
∑
i=0
(k−1
i
)
(−1)−ij
(k)
mk+i = (−2)k−1jmj
(k−1)
(m−1)(k−1);
b)
k−1
∑
i=0
(k−1
i
)
2k−i−1j
(k)
mk+i = jmj
(k−1)
(m+2)(k−1);
c)
k−1
∑
i=0
j
(k)
mk+i = jm
2jm−1
(
j
(k)
(m+1)k − j
(k)
mk
)
.
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 47
2. Generating matrices
In [15] the authors use a matrix method for generating the Jacobsthal
numbers by defining the Jacobsthal A-matrix
A =
[
1 2
1 0
]
and they proved that
An =
[
Jn+1 2Jn
Jn 2Jn−1
]
= A(n−1)
[
1 2
1 0
]
,
for any natural number n.
Thus, for any n > 0, s > 0 and n + s > 2, we have
[
Jn+s 2Jn+s−1
Jn+s−1 2Jn+s−2
]
= A(n+s−2)
[
1 2
1 0
]
.
If we compute the determinant of both sides of the previous equality
we obtain
2Jn+sJn+s−2 − 2 (Jn+s−1)2 = −2
∣
∣
∣A(n+s−2)
∣
∣
∣
which is equivalent to
(Jn+s−1)2 − Jn+sJn+s−2 = (−2)n+s−2.
Since, by Lemma 1 (where m = n + s − 1, k = 2 and r = 0)
J
(2)
2(n+s−1) = (Jn+s−1)2
,
we have proved the following result:
Theorem 3. If n, s > 0 and n + s > 2, then
J
(2)
2(n+s−1) − Jn+sJn+s−2 = (−2)n+s−2.
Also, by considering the generating Jacobsthal-Lucas B-matrix given
in [16] and in [8]
B =
[
5 2
1 4
]
48 New families of Jacobsthal and Jacobsthal-Lucas numbers
and proceeding in a similar way as we did for Jacobsthal numbers, we
obtain for n, s > 0 and n + s > 2,
[
jn+s 2jn+s−1
jn+s−1 2jn+s−2
]
= B(n+s−2)
[
5 2
1 4
]
.
Computing the determinant of both sides of this equality we get
2jn+sjn+s−2 − 2 (jn+s−1)2 =
(
322
)(n+s−2)
×
(
322
)
which is equivalent to
jn+sjn+s−2 − (jn+s−1)2 = 32(n+s−1)2(n+s−2).
Using Lemma 1 again (with m = n + s − 1, k = 2 and r = 0) we obtain
the following result:
Theorem 4. If n, s > 0 and n + s > 2, then
jn+sjn+s−2 − j
(2)
2(n+s−1) = 32(n+s−1)2(n+s−2).
3. A particular case
In this section we study the particular case of the sequences
{
J
(2)
n
}
n
and
{
j
(2)
n
}
n
defined by (1) and (2), respectively, with k = 2.
3.1. Recurrence relations
First we present a recurrence relation for these sequences.
Theorem 5. The sequences
{
J
(2)
n
}
n
and
{
j
(2)
n
}
n
satisfy, respectively, the
following recurrence relations:
J (2)
n = J
(2)
n−1 + 2J
(2)
n−3 + 4J
(2)
n−4, n = 4, 5, . . .
and
j(2)
n = j
(2)
n−1 + 2j
(2)
n−3 + 4j
(2)
n−4, n = 4, 5, . . .
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 49
Proof. First, we consider n even, that is n = 2m, for any natural number
m. In this case, using Lemma 1, we have
J
(2)
2m = (Jm)2 = JmJm
= Jm (Jm−1 + 2Jm−2)
= JmJm−1 + 2JmJm−2
= JmJm−1 + 2 (Jm−1 + 2Jm−2) Jm−2
= J
(2)
2m−1 + 2Jm−1Jm−2 + 4(Jm−2)2
= J
(2)
2m−1 + 2J
(2)
2m−3 + 4(Jm−2)2
= J
(2)
2m−1 + 2J
(2)
2m−3 + 4J
(2)
2m−4
as required. Now, for n odd, that is n = 2m + 1, for any natural number
m and, using again Lemma 1, we obtain:
J
(2)
2m+1 = JmJm+1
= Jm (Jm + 2Jm−1)
= (Jm)2 + 2Jm−1Jm
= J
(2)
2m + 2Jm−1(Jm−1 + 2Jm−2)
= J
(2)
2m + 2(Jm−1)2 + 4Jm−2Jm−1
= J
(2)
2m + 2J
(2)
2m−2 + 4Jm−2Jm−1
= J
(2)
2m + 2J
(2)
2m−2 + 4J
(2)
2m−3.
So for every n = 4, 5, . . . the result is true. In a similar way we can prove
the result for j
(2)
n .
We also note that if we consider separately the even and the odd
terms of the above defined sequences we can obtain shorter recurrence
relations. In fact, for n = 2m, for any natural number m, by Theorem 3
(with n = m and s = 1) we have
J
(2)
2m − Jm+1Jm−1 = (−2)m−1
and so
J
(2)
2m = Jm−1Jm+1 + (−2)m−1
= Jm−1(Jm + 2Jm−1) + (−2)m−1
= Jm−1Jm + 2(Jm−1)2 + (−2)m−1
= J
(2)
2m−1 + 2J
(2)
2m−2 + (−2)m−1.
50 New families of Jacobsthal and Jacobsthal-Lucas numbers
In a similar way, if we consider n = 2m + 1, for any natural number m,
we have J
(2)
2m+1 = JmJm+1 that is equal to
Jm (Jm + 2Jm−1) = (Jm)2 + 2Jm−1Jm = J
(2)
2m + 2J
(2)
2m−1.
Hence, in this case, we can conclude that
J
(2)
2m+1 = J
(2)
2m + 2J
(2)
2m−1.
Therefore we can conclude the following:
Proposition 3. A shorter recurrence relation for the sequence
{
J
(2)
n
}
n
is given by
J
(2)
2m = J
(2)
2m−1 + 2J
(2)
2m−2 + (−2)m−1
J
(2)
2m+1 = J
(2)
2m + 2J
(2)
2m−1
for the even and the odd terms.
In a similar way we obtain a shorter recurrence relation to
{
j
(2)
n
}
n
.
Proposition 4. A shorter recurrence relation for the sequence
{
j
(2)
n
}
n
is given by
j
(2)
2m = j
(2)
2m−1 + 2j
(2)
2m−2 − 32m2m−1
j
(2)
2m+1 = j
(2)
2m + 2j
(2)
2m−1
for the even and the odd terms.
Proof. The proof of the second identity is similar to the one in the previous
proposition. To the first identity, by Theorem 4 we have:
jm+1jm−1 − j
(2)
2m = 32m2m−1.
Hence
j
(2)
2m = jm+1jm−1 − 32m2m−1
= jm−1(jm + 2jm−1) − 32m2m−1
= jm−1jm + 2(jm−1)2 − 32m2m−1
= j
(2)
2m−1 + 2j
(2)
2m−2 − 32m2m−1.
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 51
3.2. Generating Functions
Next we find generating functions for these sequences. Let us suppose
that the terms of the sequences
{
J
(2)
n
}
n
and
{
j
(2)
n
}
n
are the coefficients
of a power series centred at the origin, that is convergent in
]
− 1
r1
, 1
r1
[
,
according the Proposition 2.5 in [14] and [2], respectively, for k = 2.
For
{
J
(2)
n
}
n
we obtain the following result:
Theorem 6. The generating function f (2)(x) for J
(2)
n is given by
f (2)(x) =
x2 + 2x3
1 − x − 2x3 − 4x4
.
Proof. To the sum of this power series,
f (2)(x) =
∞
∑
n=0
J (2)
n xn,
we call generating function of the generalized Jacobsthal sequence of
numbers
{
J
(2)
n
}
n
.
Then
f (2)(x) − xf (2)(x) − 2x3f (2)(x) − 4x4f (2)(x)
is equal to
(
J
(2)
0 + J
(2)
1 x + J
(2)
2 x2 + J
(2)
3 x3
)
−
(
J
(2)
0 x − J
(2)
1 x2 − J
(2)
2 x3
)
− 2J
(2)
0 x3 +
∞
∑
n=4
(
J (2)
n − J
(2)
n−1 − 2J
(2)
n−3 − 4J
(2)
n−4
)
xn.
Hence, taking into account the initial conditions of the sequence
{
J
(2)
n
}
n
,
we have
(
1 − x − 2x3 − 4x4
)
f (2)(x) =
(
0 + 0x + x2 + x3
)
−
(
0x − 0x2 − x3
)
− 2 × 0x3 +
∞
∑
n=4
(
J (2)
n −
(
J
(2)
n−1 + 2J
(2)
n−3 + 4J
(2)
n−4
))
xn.
Now, by Theorem 5, this is equivalent to
(
1 − x − 2x3 − 4x4
)
f (2)(x) = x2 + 2x3 +
∞
∑
n=4
(
J (2)
n − J (2)
n
)
and therefore
f (2)(x) =
x2 + 2x3
1 − x − 2x3 − 4x4
.
52 New families of Jacobsthal and Jacobsthal-Lucas numbers
Theorem 7. The generating function g(2)(x) for j
(2)
n is given by
g(2)(x) =
4 − 2x + 3x2 − 2x3
1 − x − 2x3 − 4x4
.
Proof. To the sum of this power series,
g(2)(x) =
∞
∑
n=0
j(2)
n xn
we call generating function of the generalized Jacobsthal-Lucas sequence
of numbers
{
j
(2)
n
}
n
.
Then, in a similar way as in the proof of the previous theorem, we
obtain
(
1 − x − 2x3 − 4x4
)
g(2)(x) =
(
j
(2)
0 + j
(2)
1 x + j
(2)
2 x2 + j
(2)
3 x3
)
−
(
j
(2)
0 x − j
(2)
1 x2 − j
(2)
2 x3
)
− 2j
(2)
0 x3
+
∞
∑
n=4
(
j(2)
n − j
(2)
n−1 − 2j
(2)
n−3 − 4j
(2)
n−4
)
xn.
Taking into account the initial conditions of the sequence
{
j
(2)
n
}
n
, we have
(
1 − x − 2x3 − 4x4
)
g(2)(x) =
(
4 + 2x + x2 + 5x3
)
−
(
4x − 2x2 − x3
)
− 8x3 +
∞
∑
n=4
(
j(2)
n −
(
j
(2)
n−1 + 2j
(2)
n−3 + 4j
(2)
n−4
))
xn.
Now, by Theorem 5, this is equivalent to
(
1 − x − 2x3 − 4x4
)
g(2)(x) = 4 − 2x + 3x2 − 2x3 +
∞
∑
n=4
(
j(2)
n − j(2)
n
)
xn
and therefore
g(2)(x) =
4 − 2x + 3x2 − 2x3
1 − x − 2x3 − 4x4
.
4. Conclusion
In this paper we have presented new families of sequences, J
(k)
n and
j
(k)
n , that generalize the Jacobsthal and the Jacobsthal-Lucas sequences
and we have established some identities involving them.
P. Catarino, P. Vasco, H. Campos, A. P. Aires, A. Borges 53
We also gave generating functions for generalized Jacobsthal and
Jacobsthal-Lucas sequences
{
J
(2)
n
}
n
and
{
j
(2)
n
}
n
.
When we were looking for more elements of these new families we have
found, first, that these families were not in the Encyclopedia of Integer
Sequences [21]. Furthermore, we have found some interesting regularities,
stated in Propositions 1 and 2.
Acknowledgements
This work has been supported by the Portuguese Government through
the FCT - Fundação para a Ciência e a Tecnologia - under the project
PEst-OE/MAT/ UI4080/2014.
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Contact information
P. Catarino,
P. Vasco,
H. Campos,
A. P. Aires,
A. Borges
Universidade de Trás-os-Montes e Alto Douro,
UTAD, Quinta de Prados, 5001-801 Vila Real,
Portugal
E-Mail(s): pcatarin@utad.pt,
pvasco@utad.pt,
hcampos@utad.pt,
aaires@utad.pt,
aborges@utad.pt
Web-page(s): http://www.utad.pt
Received by the editors: 18.11.2014
and in final form 31.12.2014.
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