On K-Fibonacci Numbers of Arithmetic Indexes

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 article info abstract

Keywords: In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic k-Fibonacci numbers sequence, say an þ r for ﬁxed integers a and r. This enables us to give in a straightforward Sequences of partial sums 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;n1 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 pffiffiffiffiffiffiffiffi n n 2 2 r1r2 kþ k þ4 k k þ4 [Binet’s formula] Fk;n ¼ r r , where r1 ¼ 2 and r2 ¼ 2 . These roots verify r1 þ r2 ¼ k, and r1 r2 ¼1 1 2 2 nþ1r 2 [Catalan’s identity] Fk;nrFk;nþr Fk;n ¼ð1Þ Fk;r 2 n [Simson’s identity] Fk;n1Fk;nþ1 Fk;n ¼ð1Þ n [D’Ocagne’s identity] Fk;mFk;nþ1 Fk;mþ1Fk;n ¼ð1Þ Fk;mn

[Convolution Product] Fk;nþm ¼ Fk;nþ1Fk;m þ Fk;nFk;m1

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 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):

n n r1 þ r2 ¼ Fk;nþ1 þ Fk;n1: ð1Þ

* Corresponding author. E-mail address: [email protected] (S. Falcon).

Proof. Applying Binet’s formula and taking into account that r1r2 ¼1 1 nþ1 nþ1 n1 n1 1 n 1 n 1 Fk;nþ1 þ Fk;n1 ¼ ðr1 r2 þ r1 r2 Þ¼ r1 r1 þ r2 r2 þ r1 r2 r1 r2 r1 r2 1 n n n n ¼ ðr1ðr1 r2Þþr2ðr1 r2ÞÞ ¼ r1 þ r2: r1 r2

a Lemma 3. Fk;aðnþ2Þþr ¼ðFk;a1 þ Fk;aþ1ÞFk;aðnþ1Þþr ð1Þ Fk;anþr

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

aðnþ1Þþr aðnþ1Þþr a a r1 r2 1 aðnþ2Þþr aðnþ2Þþr a anþr a anþr ðFk;a1 þ Fk;aþ1ÞFk;aðnþ1Þþr ¼ðr1 þ r2Þ ¼ ðr1 r2 þð1Þ r1 ð1Þ r2 Þ r1 r2 r1 r2 a ¼ Fk;aðnþ2Þþr þð1Þ Fk;anþr:

Let us denote Fk;n1 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 ð1Þ Fk;anþr: ð2Þ

1 Eq. (2) gives the general term of the k-Fibonacci sequence fFk;anþrgn¼0 as a linear combination of the two preceding terms. 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 n1 n2 ½X2 ½X2 @ ðaþ1Þi n12i n 1 i A n12i @ ðaþ1Þðiþ1Þ n22i n 2 i A n2i Fk;anþr ¼ ð1Þ Lk;a Fk;aþr þ ð1Þ Lk;a Fk;r : i¼0 i i¼0 i

In this way, the general term of sequence fFk;anþrg is written in function of the two ﬁrst terms. In particular, for a ¼ 1itis r ¼ 0, see [12], we have

n1 ½X2 n12i n 1 i Fk;n ¼ k : i¼0 i

2.1. Generating function of the sequence fFk;anþrg

Let fa;rðk; xÞ be the generating function of the sequence fFk;anþrg, with 0 6 r 6 a 1. That is, fa;rðk; xÞ¼Fk;rþ 2 Fk;aþrx þ Fk;2aþrx þ. After some easy algebra X ÀÁ a 2 a n ð1 Lk;ax þð1Þ x Þfa;rðk; xÞ¼Fk;r þðFk;aþr Fk;rLk;aÞx þ Fk;aðnþ2Þþr Lk;aFk;aðnþ1Þþr þð1Þ Fk;anþr x : 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;r1Fk;a,soFk;aþr Fk;rLk;a ¼

Fk;aFk;rþ1 Fk;aþ1Fk;r. r Finally, Fk;ar ¼ Fk;rFk;aþ1 þ Fk;r1Fk;a ¼ð1Þ ðFk;aþ1Fk;r þ Fk;aFk;rþ1Þ, and the generating function for the initial power series is

F þð1ÞrF x f ðk; xÞ¼ k;r k;ar : ð3Þ a;r a 2 1 Lk;ax þð1Þ x

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 xÞ¼1kxx2 [12,15] (2) a ¼ 2: kx (a) r ¼ 0: f2;0ðk; xÞ¼ 2 1ðk þ2Þxþx2 1x (b) r ¼ 1: f2;1ðk; xÞ¼ 2 1ðk þ2Þxþx2 182 S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185

(3) a ¼ 3: ðk2þ1Þx (a) r ¼ 0: f3;0ðk; xÞ¼ 3 1ðk þ3kÞxx2 1kx (b) r ¼ 1: f3;1ðk; xÞ¼ 3 1ðk þ3kÞxx2 kþx (c) r ¼ 2: f3;2ðk; xÞ¼ 1ðk3þ3kÞxx2

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 1. Theorem 4. Sum of the k-Fibonacci numbers of kind an þ r

Xn a r Fk;aðnþ1Þþr ð1Þ Fk;anþr Fk;r ð1Þ Fk;ar Fk;aiþr ¼ a : ð4Þ i¼0 Fk;aþ1 þ Fk;a1 ð1Þ 1

P n Proof. Applying Binnet’s formula to Sk;anþr ¼ i¼0Fk;aiþr, we get ! Xn Xn Xn raiþr raiþr 1 1 ranþrþa rr ranþrþa rr S ¼ 1 2 ¼ raiþr raiþr ¼ 1 1 2 2 k;anþr r r r r 1 2 r r ra 1 ra 1 i¼0 1 2 1 2 i¼0 i¼0 1 2 1 2 1 1 anþr a r a aðnþ1Þþr r anþr a a r aðnþ1Þþr r ¼ a a a r1 ðr1r2Þ r1r2 r1 þ r1 r2 ðr1r2Þ þ r1r2 þ r2 r2 ðr r Þ r r þ 1 r1 r2 1 2 1 2 ! 1 ranþr ranþr raðnþ1Þþr raðnþ1Þþr rr rr ra ðra ðr Þr r Þr 1 a 1 2 1 2 1 2 2 1 1 2 ¼ a a a ð Þ þ þ ð1Þ ðr1 þ r2Þþ1 r1 r2 r1 r2 r1 r2 r1 r2 a r Fk;aðnþ1Þþr ð1Þ Fk;anþr Fk;r ð1Þ Fk;ar ¼ a ; Fk;aþ1 þ Fk;a1 ð1Þ 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 Xn F þ F F ð1ÞrF F ¼ k;ð2pþ1Þðnþ1Þþr k;ð2pþ1Þnþr k;r k;ð2pþ1Þr : ð5Þ k;ð2pþ1Þiþr F þ F i¼0 k;2pþ2 k;2p For example P n Fk;nþ1þFk;n Fk;0Fk;1 Fk;nþ1þFk;n1 (1) If p ¼ 0 then a ¼ 1 ! r ¼ 0, and Fk;i ¼ ¼ [11,12] i¼0 Fk;2þFk;0 k (a) For k ¼ 1, for the classical Fibonacci sequence it is

Xn F þ F 1 F ¼ nþ1 n ¼ F 1: i k nþ2 i¼0

P n Pnþ1þPn1 (b) For k ¼ 2, for the Pell sequence we obtain i¼0Pi ¼ 2 P r n Fk;3ðnþ1Þþr þFk;3nþr Fk;r ð1Þ Fk;3r (2) If p ¼ 1 ! a ¼ 3, then Fk;3iþr ¼ 3 i¼0 k þ3k P 2 n Fk;3nþ3þFk;3nk 1 (a) r ¼ 0: Fk;3i ¼ 3 i¼0 k þ3k For the classical Fibonacci sequence, k ¼ 1, it is Xn F þ F 2 F ¼ 3nþ3 3n : 3i 4 i¼0

P n Fk;3nþ4þFk;3nþ1þk1 (b) r ¼ 1: Fk;3iþ1 ¼ 3 i¼0 k þ3k For the classical Fibonacci sequence, k ¼ 1, it is Xn F þ F F ¼ 3nþ4 3nþ1 : 3iþ1 4 i¼0 S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185 183

P n Fk;3nþ5þFk;3nþ2k1 (c) r ¼ 2: Fk;3iþ2 ¼ 3 i¼0 k þ3k For the classical Fibonacci sequence, k ¼ 1, it is

Xn F þ F 2 F ¼ 3nþ5 3nþ2 3iþ2 4 i¼0

(3) If p ¼ 2 ! a ¼ 5, then

Xn r Fk;5ðnþ1Þþr þ Fk;5nþr Fk;r ð1Þ Fk;5r Fk;5iþr ¼ 5 3 : i¼0 k þ 5k þ 5k

P 4 2 n Fk;5nþ5þFk;5nk 3k 1 (a) r ¼ 0: Fk;5i ¼ 5 3 i¼0 k þ5k þ5k P 3 n Fk;5nþ6þFk;5nþ1þk þ2k1 (b) r ¼ 1: Fk;5iþ1 ¼ 5 3 i¼0 k þ5k þ5k

P 2 n Fk;5nþ7þFk;5nþ2k k1 (c) r ¼ 2: Fk;5iþ2 ¼ 5 3 i¼0 k þ5k þ5k P 2 n Fk;5nþ8þFk;5nþ3k þk1 (d) r ¼ 3: Fk;5iþ3 ¼ 5 3 i¼0 k þ5k þ5k P 3 n Fk;5nþ9þFk;5nþ4k 2k1 (e) r ¼ 4: Fk;5iþ4 ¼ 5 3 i¼0 k þ5k þ5k

Corollary 6. Sum of even k-Fibonacci numbers If a ¼ 2p then Eq. (4) is

Xn F F F ð1ÞrF F ¼ k;2pðnþ1Þþr k;2pnþr k;r k;2pr : ð6Þ k;2piþr F þ F 2 i¼0 k;2pþ1 k;2p1 For example,

(1) If p ¼ 1 ! a ¼ 2, then

Xn r kFk;2nþ1þr Fk;r ð1Þ Fk;2r Fk;2iþr ¼ 2 : i¼0 k

P n Fk;2nþ11 (a) r ¼ 0: i¼0Fk;2i ¼ k For the classical Fibonacci sequence, k ¼ 1, it is Xn F2i ¼ F2nþ1 1: i¼0

P n Fk;2nþ2 (b) r ¼ 1: i¼0Fk;2iþ1 ¼ k For the classical Fibonacci sequence, k ¼ 1, it is

Xn F2iþ1 ¼ F2nþ2: i¼0

(2) If p ¼ 2 ! a ¼ 4, then

Xn r Fk;4ðnþ1Þþr Fk;4nþr Fk;r ð1Þ Fk;4r Fk;4iþr ¼ 4 2 : i¼0 k þ 4k

P 3 n Fk;4nþ4Fk;4nk 2k (a) r ¼ 0: Fk;4i ¼ 4 2 i¼0 k þ4k P 2 n Fk;4nþ5Fk;4nþ1þk (b) r ¼ 1: i¼0Fk;4iþ1 ¼ 4 2 P k þ4k n Fk;4nþ6Fk;4nþ22k (c) r ¼ 2: Fk;4iþ2 ¼ 4 2 i¼0 k þ4k P 2 n Fk;4nþ7Fk;4nþ3þk (d) r ¼ 3: Fk;4iþ3 ¼ 4 2 i¼0 k þ4k 184 S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185

2.3. Recurrence law for the sequence of sums of k-Fibonacci numbers of arithmetic indexes ÈÉP n It is relatively easy to prove by induction that the sequence fSk;anþrg¼ i¼0Fk;aiþr , veriﬁes the recurrence relation aþ1 r Sk;aðnþ1Þþr ¼ Lk;aSk;anþr þð1Þ Sk;aðn1Þþr þ Fk;r þð1Þ Fk;ar.

Xn Xn Xn1 Xn2 a a Sk;anþr ¼ Fk;aiþr ¼ Fk;r þ Fk;aþr þ Lk;aFk;aði1Þþr ð1Þ Fk;aði2Þþr ¼ Fk;r þ Fk;aþr þ Lk;a þFk;aiþr ð1Þ þFk;aiþr i¼0 i¼2 i¼1 i¼0 a ¼ Fk;r þ Fk;aþr þ Lk;aðSk;aðn1Þþr Fk;rÞð1Þ Sk;aðn2Þþr:

Now considering Sk;anþr and Sk;aðnþ1Þþr: a Sk;anþr ¼ð1 Lk;aÞFk;r þ Fk;aþr þ Lk;aSk;aðn1Þþr ð1Þ Sk;aðn2Þþr; a Sk;aðnþ1Þþr ¼ð1 Lk;aÞFk;r þ Fk;aþr þ Lk;aSk;anþr ð1Þ Sk;aðn1Þþr by eliminating the terms ð1 Lk;aÞFk;r þ Fk;aþr, it is deduced: a a Sk;aðnþ1Þþr ¼ð1 þ Lk;aÞSk;anþr ðLk;a þð1Þ ÞSk;aðn1Þþr þð1Þ Sk;aðn2Þþr: So, the characteristic polynomial of sequence fS g is r3 ¼ð1 þ L Þr2 ðL þð1ÞaÞr þð1ÞaS , with roots r ¼ 1, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k;anþr k;a k;a k;aðn2Þþr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 2 a 2 a Lk;aþ Lk;a4ð1Þ Lk;a Lk;a4ð1Þ a 2 a r1 ¼ 2 , and r2 ¼ 2 . These numbers verify r1 þ r2 ¼ Lk;a, r1 r2 ¼ð1Þ , and r1 r2 ¼ Lk;a 4ð1Þ . n n Then, the solution for Sk;anþr is of the form Sk;anþr ¼ C0 þ C1r1 þ C2r2. Having in mind the relations between r1 and r2, after some algebra is obtained

ðr1 Lk;aÞFk;aþr þðLk;a 1ÞFk;r C0 ¼ a ; Lk;a 1 ð1Þ a ðr1 Lk;aÞFk;aþr þð1Þ Fk;r C1 ¼ ; ðr1 r2Þðr2 1Þ a ðr2 Lk;aÞFk;aþr þð1Þ Fk;r C2 ¼ : ðr1 r2Þðr1 1Þ

Observe that, in the case a ¼ 1, r ¼ 0, and then the recurrence becomes Snþ1 ¼ð1 þ kÞSn ðk 1ÞSn1 Sn2, which, for the classical Fibonacci (that is k ¼ 1) reports Snþ1 ¼ 2Sn Sn2. n Let us now consider the alternating sequence fð1Þ Fk;an rg. In a similar way that in the preceding case, we can ﬁnd that þ r Fk;r ð1Þ Fk;ar x the generating function for this alternating sequence is ga;rðk; xÞ¼ a 2 . Moreover, the following result is given: 1Lk;axþð1Þ x Theorem 7. Alternating sum of the k-Fibonacci numbers of order an þ r Xn n nþa rþ1 i ð1Þ Fk;aðnþ1Þþr þð1Þ Fk;anþr þð1Þ Fk;ar þ Fk;r ð1Þ Fk;aiþr ¼ a i¼0 Fk;aþ1 þ Fk;a1 þð1Þ þ 1 which for different values of a and r reads as

P n n n i ð1Þ Fk;nþ1ð1Þ Fk;n 1 (1) ð1Þ Fk;i ¼ Pi¼0 n k n n i ð1Þ Fk;2nþ2þð1Þ Fk;2nk n (2) i¼0ð1Þ Fk;2i ¼ 2 ¼ð1Þ Fk;nFk;nþ1 P k þ4 n i n 2 (3) i¼0ð1Þ Fk;2iþ1 ¼ð1Þ Fk;nþ1 P n n i ð1Þ Fk;4nþ2k (4) ð1Þ Fk;4i ¼ 2 i¼0 k þ2 P n n i ð1Þ Fk;4nþ3þ1 (5) ð1Þ Fk;4iþ1 ¼ 2 i¼0 k þ2 P n n i ð1Þ Fk;4nþ4 (6) ð1Þ Fk;4iþ2 ¼ 2 i¼0 k þ2 P n n i ð1Þ Fk;4nþ5þ1 (7) ð1Þ Fk;4iþ3 ¼ 2 i¼0 k þ2

3. Conclusions

We have studied the subsequences of k-Fibonacci numbers with indexes in an arithmetic sequence. In a compact and di- rect way many formulas for the sums of such numbers have been deduced.

Acknowledgement

This work has been supported in part by CICYT Project No. MTM2008-05866-C03-02/MTM from Ministerio de Educación y Ciencia of Spain. S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185 185

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