Applied Mathematics and Computation 208 (2009) 180–185 Contents lists available at ScienceDirect 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;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 pffiffiffiffiffiffiffiffi n n 2 2 r1Àr2 kþ k þ4 kÀ k þ4 [Binet’s formula] Fk;n ¼ r Àr , where r1 ¼ 2 and r2 ¼ 2 . These roots verify r1 þ r2 ¼ k, and r1 Á r2 ¼1 1 2 2 nþ1Àr 2 [Catalan’s identity] Fk;nÀrFk;nþr À Fk;n ¼ðÀ1Þ Fk;r 2 n [Simson’s identity] Fk;nÀ1Fk;nþ1 À Fk;n ¼ðÀ1Þ n [D’Ocagne’s identity] Fk;mFk;nþ1 À Fk;mþ1Fk;n ¼ðÀ1Þ 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): n n r1 þ r2 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 ¼1 1 nþ1 nþ1 nÀ1 nÀ1 1 n 1 n 1 Fk;nþ1 þ Fk;nÀ1 ¼ ðr1 À r2 þ r1 À r2 Þ¼ r1 r1 þ À r2 r2 þ r1 À r2 r1 À r2 r1 r2 1 n n n n à ¼ ðr1ðr1 À r2Þþr2ðr1 À r2ÞÞ ¼ r1 þ r2: r1 À r2 a Lemma 3. Fk;aðnþ2Þþr ¼ðFk;aÀ1 þ Fk;aþ1ÞFk;aðnþ1Þþr ðÀ1Þ Fk;anþr Proof. Taking into account Lemma 2 and Binet’s formula: aðnþ1Þþr aðnþ1Þþr a a r1 À r2 1 aðnþ2Þþr aðnþ2Þþr a anþr a anþr ðFk;aÀ1 þ Fk;aþ1ÞFk;aðnþ1Þþr ¼ðr1 þ r2Þ ¼ ðr1 À r2 þðÀ1Þ r1 ðÀ1Þ r2 Þ r1 À r2 r1 À r2 a ¼ Fk;aðnþ2Þþr þðÀ1Þ Fk;anþr: à 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 ðÀ1Þ 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 nÀ1 nÀ2 ½X2 ½X2 @ ðaþ1Þi nÀ1À2i n À 1 À i A nÀ1À2i @ ðaþ1Þðiþ1Þ nÀ2À2i n À 2 À i A nÀ2Ài 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 nÀ1 ½X2 nÀ1À2i n À 1 À 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 À 1. That is, fa;rðk; xÞ¼Fk;rþ 2 Fk;aþrx þ Fk;2aþrx þÁÁ. After some easy algebra X ÀÁ a 2 a n ð1 À Lk;ax þðÀ1Þ x Þfa;rðk; xÞ¼Fk;r þðFk;aþr À Fk;rLk;aÞx þ Fk;aðnþ2Þþr À Lk;aFk;aðnþ1Þþr þðÀ1Þ 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;rÀ1Fk;a,soFk;aþr À Fk;rLk;a ¼ Fk;aFk;rþ1 À Fk;aþ1Fk;r. r Finally, Fk;aÀr ¼ Fk;ÀrFk;aþ1 þ Fk;ÀrÀ1Fk;a ¼ðÀ1Þ ðFk;aþ1Fk;r þ Fk;aFk;rþ1Þ, and the generating function for the initial power ser- ies is F þðÀ1ÞrF x f ðk; xÞ¼ k;r k;aÀr : ð3Þ a;r a 2 1 À Lk;ax þðÀ1Þ 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 xÞ¼1ÀkxÀx2 [12,15] (2) a ¼ 2: kx (a) r ¼ 0: f2;0ðk; xÞ¼ 2 1ðk þ2Þxþx2 1Àx (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 þ3kÞxÀx2 1Àkx (b) r ¼ 1: f3;1ðk; xÞ¼ 3 1ðk þ3kÞxÀx2 kþx (c) r ¼ 2: f3;2ðk; xÞ¼ 1ðk3þ3kÞxÀx2 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 Xn a r Fk;aðnþ1Þþr ðÀ1Þ Fk;anþr À Fk;r ðÀ1Þ Fk;aÀr Fk;aiþr ¼ a : ð4Þ i¼0 Fk;aþ1 þ Fk;aÀ1 ðÀ1Þ À 1 P n Proof. Applying Binnet’s formula to Sk;anþr ¼ i¼0Fk;aiþr, we get ! Xn Xn Xn raiþr À raiþr 1 1 ranþrþa À rr ranþrþa À rr S ¼ 1 2 ¼ raiþr À raiþr ¼ 1 1 À 2 2 k;anþr r À r r À r 1 2 r À r ra À 1 ra À 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 À r1r2 À r1 þ r1 À r2 ðr1r2Þ þ r1r2 þ r2 À r2 ðr r Þ À r À r þ 1 r1 À r2 1 2 1 2 ! 1 ranþr À ranþr raðnþ1Þþr À raðnþ1Þþr rr À rr Àra ðra ðr ÞÀr À r ÞÀr 1 a 1 2 1 2 1 2 2 1 1 2 ¼ a a a ð Þ À þ þ ð1Þ ðr1 þ r2Þþ1 r1 À r2 r1 À r2 r1 À r2 r1 À r2 a r Fk;aðnþ1Þþr ðÀ1Þ Fk;anþr À Fk;r ðÀ1Þ Fk;aÀr ¼ a ; Fk;aþ1 þ Fk;aÀ1 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 Xn F þ F À F ðÀ1ÞrF 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 ÀFk;0ÀFk;1 Fk;nþ1þFk;nÀ1 (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 À 1 F ¼ nþ1 n ¼ F À 1: i k nþ2 i¼0 P n Pnþ1þPnÀ1 (b) For k ¼ 2, for the Pell sequence we obtain i¼0Pi ¼ 2 P r n Fk;3ðnþ1Þþr þFk;3nþr ÀFk;r ð1Þ Fk;3Àr (2) If p ¼ 1 ! a ¼ 3, then Fk;3iþr ¼ 3 i¼0 k þ3k P 2 n Fk;3nþ3þFk;3nÀk À1 (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 À 2 F ¼ 3nþ3 3n : 3i 4 i¼0 P n Fk;3nþ4þFk;3nþ1þkÀ1 (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.