Applied Mathematics and Computation 208 (2009) 180–185
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
On k-Fibonacci numbers of arithmetic indexes
*
Sergio Falcon , Angel Plaza
Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), Campus de Tafira, 35017 Las Palmas de Gran Canaria, Spain
- a r t i c l e i n f o
- a b s t r a c t
Keywords:
k-Fibonacci numbers Sequences of partial sums
In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a and r. This enables us to give in a straightforward way several formulas for the sums of such numbers.
Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction
One of the more studied sequences is the Fibonacci sequence [1–3], and it has been generalized in many ways [4–10].
Here, we use the following one-parameter generalization of the Fibonacci sequence.
Definition 1. For any integer number k P 1, the kth Fibonacci sequence, say fFk;ngn2N is defined recurrently by
Fk;0 ¼ 0; Fk;1 ¼ 1; and Fk;nþ1 ¼ kFk;n þ Fk;nÀ1 for n P 1:
Note that for k ¼ 1 the classical Fibonacci sequence is obtained while for k ¼ 2 we obtain the Pell sequence. Some of the properties that the k-Fibonacci numbers verify and that we will need later are summarized below [11–15]:
pffiffiffiffiffiffiffiffi
kþ k2þ4
2
pffiffiffiffiffiffiffiffi
rn rn2
À
kÀ k2
þ4. These roots verify r1
þ
r2 ¼ k, and r1 r2 ¼ À1
Á
r11Àr2
2
ꢀ [Binet’s formula] Fk;n
¼
, where r1
¼
and r2
¼
ꢀ [Catalan’s identity] Fk;nÀrFk;nþr À F2 ¼ ðÀ1Þnþ1ÀrFk2;r
k;n n
ꢀ [Simson’s identity] Fk;nÀ1Fk;nþ1 À F2 ¼ ðÀ1Þ
k;n n
ꢀ [D’Ocagne’s identity] Fk;mFk;nþ1 À Fk;mþ1Fk;n ¼ ðÀ1Þ Fk;mÀn ꢀ [Convolution Product] Fk;nþm ¼ Fk;nþ1Fk;m þ Fk;nFk;mÀ1
In this paper, we study different sums of k-Fibonacci numbers. Sums of Fibonacci numbers appear in different contexts, even they are related with the dimensionality of heterotic superstrings [16,17]. We focus here on the subsequences of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a, r with 0 6 r 6 a À 1. Several formulas for the sums of such numbers are deduced in a straightforward way.
2. On the k-Fibonacci numbers of kind an þ r
Let us prove two lemmas that we will need later.
Lemma 2. For all integer n (n P 1):
r1n
þ
rn2 ¼ Fk;nþ1 þ Fk;nÀ1:
ð1Þ
* Corresponding author.
E-mail address: [email protected] (S. Falcon).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.031
S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185
181
Proof. Applying Binet’s formula and taking into account that r1r2 ¼ À1
ꢀ
rn2:
- ꢀ
- ꢁ
- ꢀ
- ꢁꢁ
1
À
1
À
1
À
1
r1
1
r2
Fk;nþ1 þ Fk;nÀ1
¼¼ðð
rn1þ1
À
rn2þ1
þ
rn1À1
Àrn2À1Þ ¼
rn1 r1
- þ
- À
rn2 r2
þ
r1 r1 r2 r2
- r1
- r2
r1n
ð
r1
Àr2Þ þ rn2ðr1
Àr2ÞÞ ¼ rn1 þ
Ã
a
Lemma 3. Fk;aðnþ2Þþr ¼ ðFk;aÀ1 þ Fk;aþ1ÞFk;aðnþ1Þþr À ðÀ1Þ Fk;anþr
Proof. Taking into account Lemma 2 and Binet’s formula:
ra1ðnþ1Þþr
ÀÀ
ra2ðnþ1Þþr
r2
1
À
ra1ðnþ2Þþr
À
ra2ðnþ2Þþr þ ðÀ1Þ r1anþr À ðÀ1Þ ra2nþr
Þ
- a
- a
ðFk;aÀ1 þ Fk;aþ1ÞFk;aðnþ1Þþr ¼ ðr1a
þ
r2a
- Þ
- ¼
- ð
- r1
- r1
- r2
a
¼ Fk;aðnþ2Þþr þ ðÀ1Þ Fk;anþr
:
Ã
Let us denote Fk;nÀ1 þ Fk;nþ1 by Lk;n (numbers Lk;n are called k-Lucas numbers).
Then previous formula becomes
a
Fk;aðnþ2Þþr ¼ Lk;aFk;aðnþ1Þþr À ðÀ1Þ Fk;anþr
:
ð2Þ
Eq. (2) gives the general term of the k-Fibonacci sequence fFk;anþrg1 as a linear combination of the two preceding terms.
n¼0
Note that, applying iteratively this formula, the general term can be written as a non-linear combination of the two first terms of the sequence:
- 0
- 1
- 0
- 1
- nÀ1
- nÀ2
- ꢀ
- ꢁ
- ꢀ
- ꢁ
- ½
- ꢁ
- ½
- ꢁ
- 2
- 2
- X
- X
- n À 1 À i
- n À 2 À i
ðÀ1Þðaþ1Þi
L
Fk;aþr
þ
ðÀ1Þðaþ1Þðiþ1Þ
L
Fk;r
- nÀ1À2i
- nÀ1À2i
- nÀ2À2i
- nÀ2Ài
:
- @
- A
- @
- A
Fk;anþr
¼
- k;a
- k;a
- i
- i
- i¼0
- i¼0
In this way, the general term of sequence fFk;anþrg is written in function of the two first terms. In particular, for a ¼ 1 it is r ¼ 0, see [12], we have
nÀ1
- ꢀ
- ꢁ
- ½
- ꢁ
2
X
n À 1 À i
Fk;n ¼ knÀ1À2i
:
i
i¼0
2.1. Generating function of the sequence fFk;anþr
g
Let fa;rðk; xÞ be the generating function of the sequence fFk;anþrg, with 0 6 r 6 a À 1. That is, fa;rðk; xÞ ¼ Fk;r
þ¼
Fk;aþrx þ Fk;2aþrx2 þ Á Á Á. After some easy algebra
X
- À
- Á
- a
- a
ð1 À Lk;ax þ ðÀ1Þ x2Þfa;rðk; xÞ ¼ Fk;r þ ðFk;aþr À Fk;rLk;aÞx þ
Fk;aðnþ2Þþr À Lk;aFk;aðnþ1Þþr þ ðÀ1Þ Fk;anþr xn:
nP2
First, taking into account Lemma 3, the series of the Right Hand Side vanishes.
On the other hand, the Convolution Product Identity establishes that Fk;rþa ¼ Fk;rFk;aþ1 þ Fk;rÀ1Fk;a, so Fk;aþr À Fk;rLk;a
Fk;aFk;rþ1 À Fk;aþ1Fk;r
.
r
Finally, Fk;aÀr ¼ Fk;ÀrFk;aþ1 þ Fk;ÀrÀ1Fk;a ¼ ðÀ1Þ ðÀFk;aþ1Fk;r þ Fk;aFk;rþ1Þ, and the generating function for the initial power series is
r
Fk;r þ ðÀ1Þ Fk;aÀr
x
fa;rðk; xÞ ¼
:
ð3Þ
a
1 À Lk;ax þ ðÀ1Þ x2
2.1.1. Particular cases
The generating functions of sequences fFk;anþrg for different values of parameters a and r are
x
(1) a ¼ 1 and then r ¼ 0: f1;0ðk; xÞ ¼
[12,15]
1ÀkxÀx2
(2) a ¼ 2:
kx
(a) (b)
r ¼ 0: f2;0ðk; xÞ ¼
1Àðk2þ2Þxþx2
1Àx
r ¼ 1: f2;1ðk; xÞ ¼
1Àðk2þ2Þxþx2
182
S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185
(3) a ¼ 3:
ðk2þ1Þx
(a) (b) (c)
r ¼ 0: f3;0ðk; xÞ ¼ r ¼ 1: f3;1ðk; xÞ ¼ r ¼ 2: f3;2ðk; xÞ ¼
1Àðk3þ3kÞxÀx2
1Àkx
1Àðk3þ3kÞxÀx2
kþx
1Àðk3þ3kÞxÀx2
2.2. Sum of k-Fibonacci numbers of kind an þ r
In this section, we study the sum of the k-Fibonacci numbers of kind an þ r, with a an integer number, and
r ¼ 0; 1; 2; . . . ; a À 1.
Theorem 4. Sum of the k-Fibonacci numbers of kind an þ r
- a
- r
n
X
Fk;aðnþ1Þþr À ðÀ1Þ Fk;anþr À Fk;r À ðÀ1Þ Fk;aÀr
Fk;aiþr
¼
:
ð4Þ
a
Fk;aþ1 þ Fk;aÀ1 À ðÀ1Þ À 1
i¼0
P
n
Proof. Applying Binnet’s formula to Sk;anþr
¼
i¼0Fk;aiþr, we get
- !
- ꢀ
- ꢁ
- n
- n
- n
aiþr
X
r1
ÀÀ
ra2iþr
r2
1
À
1
À
ra1nþrþa
À
rr1 ra2nþrþa
À
r2r
- X
- X
Sk;anþr
¼¼¼¼
¼
ra1iþr
À
ra2iþr
- ¼
- À
- r1
- r1
- r2
- r1
- r2
- ra1 À 1
- ra2 À 1
- i¼0
- i¼0
- i¼0
- ꢂ
- ꢃ
- 1
- 1
- a
- a
ra1nþrðr1r2Þ À r1r ra2
À
ra1ðnþ1Þþr
þ
r1r
Àra2nþrðr1r2Þ þ r1arr2 þ
ra2ðnþ1Þþr
À
r2r
a
r1r2Þ À ra1
À
ra2 þ 1
r
1 À r2
ð
- !
1
a ranþr
1r1
ÀÀ
r
ra2nþr ra1ðnþ1Þþr
r2
ÀÀ
ra2ðnþ1Þþr rr1
ÀÀ
r2r
r2
À
r2a
ð
ra1ðÀr1ÞÀr
À
r2ÞÀr
ðÀ1Þ
- À
- þ
- þ
a
ðÀ1Þ À ðra1 þra2Þ þ 1
- r1
- r2
- r1
- r1
À
r2
a
Fk;aðnþ1Þþr À ðÀ1Þ Fk;anþr À Fk;r À ðÀ1Þ Fk;aÀr
;
a
Fk;aþ1 þ Fk;aÀ1 À ðÀ1Þ À 1
where we have used Eq. (2).
h
For k ¼ 1; 2; 3 different sequences of these partial sums are listed in OEIS [18].
Corollary 5. Sum of odd k-Fibonacci numbers
If a ¼ 2p þ 1 then Eq. (4) is
- r
- n
X
Fk;ð2pþ1Þðnþ1Þþr þ Fk;ð2pþ1Þnþr À Fk;r À ðÀ1Þ Fk;ð2pþ1ÞÀr
Fk;ð2pþ1Þiþr