On K-Fibonacci Numbers of Arithmetic Indexes

Applied Mathematics and Computation 208 (2009) 180–185 Contents lists available at <a href="/goto?url=http://www.sciencedirect.com/science/journal/00963003" target="_blank">ScienceDirect </a>Applied Mathematics and Computation journal homepage: <a href="/goto?url=http://www.elsevier.com/locate/amc" target="_blank">w</a><a href="/goto?url=http://www.elsevier.com/locate/amc" target="_blank">ww.elsevier.com/locate/amc </a>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 <ul style="display: flex;"><li style="flex:1">a r t i c l e i n f o </li><li style="flex:1">a b s t r a c t </li></ul>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 2pffiffiffiffiffiffiffiffi rn rn2 ÀkÀ k2 þ4. These roots verify r1 þr2 ¼ k, and r1 r2 ¼ À1 Ár11Àr2 2ꢀ [Binet’s formula] Fk;n ¼, where r1 ¼and r2 ¼ꢀ [Catalan’s identity] Fk;nÀrFk;nþr À F2 ¼ ðÀ1Þnþ1ÀrFk2;r k;n nꢀ [Simson’s identity] Fk;nÀ1Fk;nþ1 À F2 ¼ ðÀ1Þ k;n nꢀ [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 þrn2 ¼ Fk;nþ1 þ Fk;nÀ1: ð1Þ * Corresponding author. E-mail address: <a href="mailto:[email protected]" target="_blank">[email protected] </a>(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: <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁꢁ </li></ul>1À1À1À1r1 1r2 Fk;nþ1 þ Fk;nÀ1 ¼¼ððrn1þ1 Àrn2þ1 þrn1À1 Àrn2À1Þ ¼ rn1 r1 <ul style="display: flex;"><li style="flex:1">þ</li><li style="flex:1">À</li></ul>rn2 r2 þr1 r1 r2 r2 <ul style="display: flex;"><li style="flex:1">r1 </li><li style="flex:1">r2 </li></ul>r1n ðr1 Àr2Þ þ rn2ðr1 Àr2ÞÞ ¼ rn1 þÃaLemma 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 ÀÀra2ðnþ1Þþr r2 1Àra1ðnþ2Þþr Àra2ðnþ2Þþr þ ðÀ1Þ r1anþr À ðÀ1Þ ra2nþr Þ<ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">a</li></ul>ðFk;aÀ1 þ Fk;aþ1ÞFk;aðnþ1Þþr ¼ ðr1a þr2a <ul style="display: flex;"><li style="flex:1">Þ</li><li style="flex:1">¼</li><li style="flex:1">ð</li></ul><ul style="display: flex;"><li style="flex:1">r1 </li><li style="flex:1">r1 </li><li style="flex:1">r2 </li></ul>a¼ Fk;aðnþ2Þþr þ ðÀ1Þ Fk;anþr :ÃLet us denote Fk;nÀ1 þ Fk;nþ1 by Lk;n (numbers Lk;n are called k-Lucas numbers). Then previous formula becomes aFk;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: <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">1</li></ul><ul style="display: flex;"><li style="flex:1">nÀ1 </li><li style="flex:1">nÀ2 </li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">½</li><li style="flex:1">ꢁ</li><li style="flex:1">½</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><ul style="display: flex;"><li style="flex:1">n À 1 À i </li><li style="flex:1">n À 2 À i </li></ul>ðÀ1Þðaþ1Þi LFk;aþr þðÀ1Þðaþ1Þðiþ1Þ LFk;r <ul style="display: flex;"><li style="flex:1">nÀ1À2i </li><li style="flex:1">nÀ1À2i </li><li style="flex:1">nÀ2À2i </li><li style="flex:1">nÀ2Ài </li></ul>:<ul style="display: flex;"><li style="flex:1">@</li><li style="flex:1">A</li><li style="flex:1">@</li><li style="flex:1">A</li></ul>Fk;anþr ¼<ul style="display: flex;"><li style="flex:1">k;a </li><li style="flex:1">k;a </li></ul><ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">i</li></ul><ul style="display: flex;"><li style="flex:1">i¼0 </li><li style="flex:1">i¼0 </li></ul>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 <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">½</li><li style="flex:1">ꢁ</li></ul>2Xn À 1 À i Fk;n ¼ knÀ1À2i :ii¼0 2.1. Generating function of the sequence fFk;anþr gLet 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 þ¼Fk;aþrx þ Fk;2aþrx2 þ Á Á Á. After some easy algebra X<ul style="display: flex;"><li style="flex:1">À</li><li style="flex:1">Á</li></ul><ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">a</li></ul>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 .rFinally, 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 rFk;r þ ðÀ1Þ Fk;aÀr xfa;rðk; xÞ ¼ :ð3Þ a1 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 <ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">r</li></ul>nXFk;aðnþ1Þþr À ðÀ1Þ Fk;anþr À Fk;r À ðÀ1Þ Fk;aÀr Fk;aiþr ¼:ð4Þ aFk;aþ1 þ Fk;aÀ1 À ðÀ1Þ À 1 i¼0 PnProof. Applying Binnet’s formula to Sk;anþr ¼i¼0Fk;aiþr, we get <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li><li style="flex:1">n</li></ul>aiþr Xr1 ÀÀra2iþr r2 1À1Àra1nþrþa Àrr1 ra2nþrþa Àr2r <ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul>Sk;anþr ¼¼¼¼ ¼ra1iþr Àra2iþr <ul style="display: flex;"><li style="flex:1">¼</li><li style="flex:1">À</li></ul><ul style="display: flex;"><li style="flex:1">r1 </li><li style="flex:1">r1 </li><li style="flex:1">r2 </li><li style="flex:1">r1 </li><li style="flex:1">r2 </li><li style="flex:1">ra1 À 1 </li><li style="flex:1">ra2 À 1 </li></ul><ul style="display: flex;"><li style="flex:1">i¼0 </li><li style="flex:1">i¼0 </li><li style="flex:1">i¼0 </li></ul><ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><ul style="display: flex;"><li style="flex:1">a</li><li style="flex:1">a</li></ul>ra1nþrðr1r2Þ À r1r ra2 Àra1ðnþ1Þþr þr1r Àra2nþrðr1r2Þ þ r1arr2 þra2ðnþ1Þþr Àr2r ar1r2Þ À ra1 Àra2 þ 1 r1 À r2 ð<ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul>1a ranþr 1r1 ÀÀrra2nþr ra1ðnþ1Þþr r2 ÀÀra2ðnþ1Þþr rr1 ÀÀr2r r2 Àr2a ðra1ðÀr1ÞÀr Àr2ÞÀr ðÀ1Þ <ul style="display: flex;"><li style="flex:1">À</li><li style="flex:1">þ</li><li style="flex:1">þ</li></ul>aðÀ1Þ À ðra1 þra2Þ þ 1 <ul style="display: flex;"><li style="flex:1">r1 </li><li style="flex:1">r2 </li><li style="flex:1">r1 </li><li style="flex:1">r1 </li></ul>Àr2 aFk;aðnþ1Þþr À ðÀ1Þ Fk;anþr À Fk;r À ðÀ1Þ Fk;aÀr ;aFk;aþ1 þ Fk;aÀ1 À ðÀ1Þ À 1 where we have used Eq. (2). hFor 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 <ul style="display: flex;"><li style="flex:1">r</li><li style="flex:1">n</li></ul>XFk;ð2pþ1Þðnþ1Þþr þ Fk;ð2pþ1Þnþr À Fk;r À ðÀ1Þ Fk;ð2pþ1ÞÀr Fk;ð2pþ1Þiþr

On K-Fibonacci Numbers of Arithmetic Indexes

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