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 fF_k;ng_n2N is defined recurrently by

F_k;0 ¼ 0; F_k;1 ¼ 1; and F_k;nþ1 ¼ kF_k;n þ F_k;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] F_k;n

¼

, where r₁

¼

and r₂

¼

ꢀ [Catalan’s identity] F_k;nÀrF_k;nþr À F² ¼ ðÀ1Þ^nþ1ÀrF_k²_;r

k;n n

ꢀ [Simson’s identity] F_k;nÀ1F_k;nþ1 À F² ¼ ðÀ1Þ

k;n n

ꢀ [D’Ocagne’s identity] F_k;mF_k;nþ1 À F_k;mþ1F_k;n ¼ ðÀ1Þ F_k;mÀn ꢀ [Convolution Product] F_k;nþm ¼ F_k;nþ1F_k;m þ F_k;nF_k;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 F_k;nþ1 þ F_k;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

F_k;nþ1 þ F_k;nÀ1

¼¼ðð

rⁿ₁^þ1

À

rⁿ₂^þ1

þ

rⁿ₁^À1

Àrⁿ₂^À1Þ 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. F_{k;aðnþ2Þþr} ¼ ðF_k;aÀ1 þ F_k;aþ1ÞF_{k;aðnþ1Þþr} À ðÀ1Þ F_k;anþr

Proof. Taking into account Lemma 2 and Binet’s formula:

r^a₁^ðnþ1Þþr

ÀÀ

r^a₂^ðnþ1Þþr

r₂

1

À

r^a₁^ðnþ2Þþr

À

r^a₂^ðnþ2Þþr þ ðÀ1Þ r₁^anþr À ðÀ1Þ r^a₂^nþr

Þ

• a
• a

ðF_k;aÀ1 þ F_k;aþ1ÞF_{k;aðnþ1Þþr} ¼ ðr₁^a

þ

r₂^a

• Þ
• ¼
• ð

• r₁
• r₁
• r₂

a

¼ F_{k;aðnþ2Þþr} þ ðÀ1Þ F_k;anþr

:

Ã

Let us denote F_k;nÀ1 þ F_k;nþ1 by L_k;n (numbers L_k;n are called k-Lucas numbers).
Then previous formula becomes

a

F_{k;aðnþ2Þþr} ¼ L_k;aF_{k;aðnþ1Þþr} À ðÀ1Þ F_k;anþr

:

ð2Þ

Eq. (2) gives the general term of the k-Fibonacci sequence fF_k;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

ðÀ1Þ^ðaþ1Þi

L

F_k;aþr

þ

ðÀ1Þ^{ðaþ1Þðiþ1Þ}

L

F_k;r

• nÀ1À2i
• nÀ1À2i
• nÀ2À2i
• nÀ2Ài

:

• @
• A
• @
• A

F_k;anþr

¼

• k;a
• k;a

• i
• i

• i¼0
• i¼0

In this way, the general term of sequence fF_k;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

F_k;n ¼ k^nÀ1À2i

:

i

i¼0

2.1. Generating function of the sequence fF_k;anþr

g

Let f_a;rðk; xÞ be the generating function of the sequence fF_k;anþrg, with 0 6 r 6 a À 1. That is, f_a;rðk; xÞ ¼ F_k;r

þ¼

F_k;aþrx þ F_k;2aþrx² þ Á Á Á. After some easy algebra

X

• À
• Á

• a
• a

ð1 À L_k;ax þ ðÀ1Þ x²Þf_a;rðk; xÞ ¼ F_k;r þ ðF_k;aþr À F_k;rL_k;aÞx þ

F_{k;aðnþ2Þþr} À L_k;aF_{k;aðnþ1Þþr} þ ðÀ1Þ F_k;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 F_k;rþa ¼ F_k;rF_k;aþ1 þ F_k;rÀ1F_k;a, so F_k;aþr À F_k;rL_k;a

F_k;aF_k;rþ1 À F_k;aþ1F_k;r

.

r

Finally, F_k;aÀr ¼ F_k;ÀrF_k;aþ1 þ F_k;ÀrÀ1F_k;a ¼ ðÀ1Þ ðÀF_k;aþ1F_k;r þ F_k;aF_k;rþ1Þ, and the generating function for the initial power series is

r

F_k;r þ ðÀ1Þ F_k;aÀr

x

f_a;rðk; xÞ ¼

:

ð3Þ

a

1 À L_k;ax þ ðÀ1Þ x²

2.1.1. Particular cases

The generating functions of sequences fF_k;anþrg for different values of parameters a and r are

x

(1) a ¼ 1 and then r ¼ 0: f_1;0ðk; xÞ ¼

[12,15]

1ÀkxÀx²

(2) a ¼ 2:

kx

(a) (b)

r ¼ 0: f_2;0ðk; xÞ ¼

1Àðk²þ2Þxþx²

1Àx

r ¼ 1: f_2;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: f_3;0ðk; xÞ ¼ r ¼ 1: f_3;1ðk; xÞ ¼ r ¼ 2: f_3;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

F_{k;aðnþ1Þþr} À ðÀ1Þ F_k;anþr À F_k;r À ðÀ1Þ F_k;aÀr
F_k;aiþr

¼

:

ð4Þ

a

F_k;aþ1 þ F_k;aÀ1 À ðÀ1Þ À 1

i¼0

P

n

Proof. Applying Binnet’s formula to S_k;anþr

¼

_i¼0F_k;aiþr, we get

•  
• !

• ꢀ
• ꢁ

• n
• n
• n

aiþr

X

r₁

ÀÀ

r^a₂^iþr

r₂

1

À

1

À

r^a₁^nþrþa

À

r^r₁ r^a₂^nþrþa

À

r₂^r

• X
• X

S_k;anþr

¼¼¼¼
¼

r^a₁^iþr

À

r^a₂^iþr

• ¼
• À

• r₁
• r₁
• r₂
• r₁
• r₂
• r^a₁ À 1
• r^a₂ À 1

• i¼0
• i¼0
• i¼0

• ꢂ
• ꢃ

• 1
• 1

• a
• a

r^a₁^nþrðr₁r₂Þ À r₁^r r^a₂
À

r^a₁^ðnþ1Þþr

þ

r₁^r

Àr^a₂^nþrðr₁r₂Þ þ r₁^ar^r₂ þ

r^a₂^ðnþ1Þþr

À

r₂^r

a

r₁r₂Þ À r^a₁
À

r^a₂ þ 1

r

₁ À r₂

ð

•  
• !

1

_a r^anþr

¹r₁

ÀÀ

r

r^a₂^nþr r^a₁^ðnþ1Þþr

r₂

ÀÀ

r^a₂^ðnþ1Þþr r^r₁

ÀÀ

r₂^r

r₂

À

r₂^a

ð

r^a₁ðÀr₁Þ^Àr

À

r₂Þ^Àr

ðÀ1Þ

• À
• þ
• þ

a

ðÀ1Þ À ðr^a₁ þr^a₂Þ þ 1

• r₁
• r₂
• r₁
• r₁

À

r₂

a

F_{k;aðnþ1Þþr} À ðÀ1Þ F_k;anþr À F_k;r À ðÀ1Þ F_k;aÀr

;

a

F_k;aþ1 þ F_k;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

F_{k;ð2pþ1Þðnþ1Þþr} þ F_{k;ð2pþ1Þnþr} À F_k;r À ðÀ1Þ F_{k;ð2pþ1ÞÀr}
F_{k;ð2pþ1Þiþr}