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
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.
Keywords:
k-Fibonacci numbers
Sequences of partial sums
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;n gn2N 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;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
2
2
rn rn
[Binet’s formula] F k;n ¼ r11 r22 , where r1 ¼ kþ 2k þ4 and r2 ¼ k 2k þ4. These roots verify r1 þ r2 ¼ k, and r1 r2 ¼ 1
nþ1r 2
2
F k;r
[Catalan’s identity] F k;nr F k;nþr F k;n ¼ ð1Þ
[Simson’s identity] F k;n1 F k;nþ1 F 2k;n ¼ ð1Þn
[D’Ocagne’s identity] F k;m F k;nþ1 F k;mþ1 F k;n ¼ ð1Þn F k;mn
[Convolution Product] F k;nþm ¼ F k;nþ1 F k;m þ F k;n F k;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):
rn1 þ rn2 ¼ F k;nþ1 þ F k;n1 :
* 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
ð1Þ
S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185
Proof. Applying Binet’s formula and taking into account that
181
r1 r2 ¼ 1
1
1
1
1
nþ1
n1
n1
n
n
ðrnþ1
r
þ
r
r
Þ
¼
r
r
þ
r
r
þ
2
2
1
2
2
r1 r2 1
r1 r2 1 1 r1
r2
1
n
n
n
n
ðr ðr r2 Þ þ r2 ðr1 r2 ÞÞ ¼ r1 þ r2 :
¼
r1 r2 1 1
F k;nþ1 þ F k;n1 ¼
Lemma 3. F k;aðnþ2Þþr ¼ ðF k;a1 þ F k;aþ1 ÞF k;aðnþ1Þþr ð1Þa F k;anþr
Proof. Taking into account Lemma 2 and Binet’s formula:
ðF k;a1 þ F k;aþ1 ÞF k;aðnþ1Þþr ¼ ðra1 þ ra2 Þ
r1aðnþ1Þþr raðnþ1Þþr
1
aðnþ2Þþr
aðnþ2Þþr
2
¼
ðr
r2
þ ð1Þa r1anþr ð1Þa r2anþr Þ
r1 r2
r1 r2 1
¼ F k;aðnþ2Þþr þ ð1Þa F k;anþr :
Let us denote F k;n1 þ F k;nþ1 by Lk;n (numbers Lk;n are called k-Lucas numbers).
Then previous formula becomes
F k;aðnþ2Þþr ¼ Lk;a F k;aðnþ1Þþr ð1Þa F k;anþr :
ð2Þ
Eq. (2) gives the general term of the k-Fibonacci sequence fF k;anþr g1
n¼0 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:
F k;anþr
0 n1
1
0 n2
1
½X
½X
2 2 n1i
n2i
ðaþ1Þi
ðaþ1Þðiþ1Þ
n12i
n12i
n22i
AF
@ ð1Þ
AF n2i :
¼ @ ð1Þ
Lk;a
Lk;a
k;aþr þ
k;r
i
i
i¼0
i¼0
In this way, the general term of sequence fF k;anþr g is written in function of the two first terms. In particular, for a ¼ 1 it is
r ¼ 0, see [12], we have
n12i
F k;n ¼ k
n1 ½X
2 n1i
i¼0
i
:
2.1. Generating function of the sequence fF k;anþr g
Let fa;r ðk; xÞ be the generating function of the sequence fF k;anþr g, with 0 6 r 6 a 1. That is, fa;r ðk; xÞ ¼ F k;r þ
F k;aþr x þ F k;2aþr x2 þ . After some easy algebra
ð1 Lk;a x þ ð1Þa x2 Þfa;r ðk; xÞ ¼ F k;r þ ðF k;aþr F k;r Lk;a Þx þ
X
F k;aðnþ2Þþr Lk;a F k;aðnþ1Þþr þ ð1Þa 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;r F k;aþ1 þ F k;r1 F k;a , so F k;aþr F k;r Lk;a ¼
F k;a F k;rþ1 F k;aþ1 F k;r .
Finally, F k;ar ¼ F k;r F k;aþ1 þ F k;r1 F k;a ¼ ð1Þr ðF k;aþ1 F k;r þ F k;a F k;rþ1 Þ, and the generating function for the initial power series is
fa;r ðk; xÞ ¼
F k;r þ ð1Þr F k;ar x
:
1 Lk;a x þ ð1Þa x2
2.1.1. Particular cases
The generating functions of sequences fF k;anþr g for different values of parameters a and r are
x
(1) a ¼ 1 and then r ¼ 0: f1;0 ðk; xÞ ¼ 1kxx
2 [12,15]
(2) a ¼ 2:
kx
(a)
r ¼ 0: f2;0 ðk; xÞ ¼ 1ðk2 þ2Þxþx
2
(b)
r ¼ 1: f2;1 ðk; xÞ ¼ 1ðk21x
þ2Þxþx2
ð3Þ
182
S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185
(3) a ¼ 3:
2
þ1Þx
(a)
r ¼ 0: f3;0 ðk; xÞ ¼ 1ðkðk3 þ3kÞxx
2
(b)
r ¼ 1: f3;1 ðk; xÞ ¼ 1ðk31kx
þ3kÞxx2
(c)
r ¼ 2: f3;2 ðk; xÞ ¼ 1ðk3 kþx
þ3kÞxx2
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
n
X
F k;aiþr ¼
i¼0
F k;aðnþ1Þþr ð1Þa F k;anþr F k;r ð1Þr F k;ar
:
F k;aþ1 þ F k;a1 ð1Þa 1
Proof. Applying Binnet’s formula to Sk;anþr ¼
n
X
Sk;anþr ¼
i¼0
Pn
i¼0 F k;aiþr ,
ð4Þ
we get
!
anþrþa
n
n
X
X
raiþr
raiþr
r1
rr1 ranþrþa
rr2
1
1
1
2
2
¼
¼
r1aiþr raiþr
2
r1 r2
r1 r2 i¼0
r1 r2
ra1 1
ra2 1
i¼0
1
aðnþ1Þþr
aðnþ1Þþr
a
a
anþr
r a
r
anþr
a r
r
r
ð
r
r
Þ
r
r
r
þ
r
r
ð
r
r
Þ
þ
r
r
þ
r
r
1
2
1
2
1
1
2
1
2
1
2
2
1
2
ðr1 r2 Þ ra1 ra2 þ 1 r1 r2
!
aðnþ1Þþr
anþr
1
ranþr
raðnþ1Þþr
r2
rr1 rr2 ra2 ðra1 ðr1 Þr r2 Þr
a r1
2
1
ð1Þ
þ
þ
¼
r1 r2
r1 r2
r1 r2
r1 r2
ð1Þa ðra1 þ ra2 Þ þ 1
1
¼
a
F k;aðnþ1Þþr ð1Þa F k;anþr F k;r ð1Þr F k;ar
;
F k;aþ1 þ F k;a1 ð1Þa 1
¼
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
n
X
F k;ð2pþ1Þiþr ¼
i¼0
F k;ð2pþ1Þðnþ1Þþr þ F k;ð2pþ1Þnþr F k;r ð1Þr F k;ð2pþ1Þr
:
F k;2pþ2 þ F k;2p
ð5Þ
For example
(1) If p ¼ 0 then a ¼ 1 ! r ¼ 0, and
Pn
i¼0 F k;i
¼
F k;nþ1 þF k;n F k;0 F k;1
F k;2 þF k;0
¼
F k;nþ1 þF k;n 1
k
(a) For k ¼ 1, for the classical Fibonacci sequence it is
n
X
i¼0
Fi ¼
F nþ1 þ F n 1
¼ F nþ2 1:
k
P
n 1
(b) For k ¼ 2, for the Pell sequence we obtain ni¼0 P i ¼ Pnþ1 þP
2
r
Pn
F
þF
F k;r ð1Þ F k;3r
(2) If p ¼ 1 ! a ¼ 3, then i¼0 F k;3iþr ¼ k;3ðnþ1Þþr k;3nþr
k3 þ3k
Pn
F k;3nþ3 þF k;3n k2 1
(a)
r ¼ 0: i¼0 F k;3i ¼
k3 þ3k
For the classical Fibonacci sequence, k ¼ 1, it is
n
X
i¼0
(b)
n
X
i¼0
F 3i ¼
F 3nþ3 þ F 3n 2
:
4
P
F
þF k;3nþ1 þk1
r ¼ 1: ni¼0 F k;3iþ1 ¼ k;3nþ4 k3 þ3k
For the classical Fibonacci sequence, k ¼ 1, it is
F 3iþ1 ¼
F 3nþ4 þ F 3nþ1
:
4
[11,12]
S. Falcon, A. Plaza / Applied Mathematics and Computation 208 (2009) 180–185
(c)
n
X
183
P
F
þF k;3nþ2 k1
r ¼ 2: ni¼0 F k;3iþ2 ¼ k;3nþ5 k3 þ3k
For the classical Fibonacci sequence, k ¼ 1, it is
F 3iþ2 ¼
i¼0
F 3nþ5 þ F 3nþ2 2
4
(3) If p ¼ 2 ! a ¼ 5, then
n
X
F k;5iþr ¼
F k;5ðnþ1Þþr þ F k;5nþr F k;r ð1Þr F k;5r
5
(a)
(b)
(c)
(d)
(e)
3
k þ 5k þ 5k
i¼0
r ¼ 0:
Pn
r ¼ 1:
Pn
¼
r ¼ 2:
Pn
¼
r ¼ 3:
Pn
¼
r ¼ 4:
Pn
¼
i¼0 F k;5i
¼
i¼0 F k;5iþ1
i¼0 F k;5iþ2
i¼0 F k;5iþ3
i¼0 F k;5iþ4
:
F k;5nþ5 þF k;5n k4 3k2 1
k5 þ5k3 þ5k
F k;5nþ6 þF k;5nþ1 þk3 þ2k1
k5 þ5k3 þ5k
F k;5nþ7 þF k;5nþ2 k2 k1
k5 þ5k3 þ5k
F k;5nþ8 þF k;5nþ3 k2 þk1
k5 þ5k3 þ5k
F k;5nþ9 þF k;5nþ4 k3 2k1
k5 þ5k3 þ5k
Corollary 6. Sum of even k-Fibonacci numbers
If a ¼ 2p then Eq. (4) is
n
X
F k;2piþr ¼
i¼0
F k;2pðnþ1Þþr F k;2pnþr F k;r ð1Þr F k;2pr
:
F k;2pþ1 þ F k;2p1 2
For example,
(1) If p ¼ 1 ! a ¼ 2, then
n
X
F k;2iþr ¼
kF k;2nþ1þr F k;r ð1Þr F k;2r
k
i¼0
(a)
n
X
r ¼ 0:
Pn
i¼0 F k;2i
¼
2
F k;2nþ1 1
For
k
:
the classical Fibonacci sequence, k ¼ 1, it is
F 2i ¼ F 2nþ1 1:
i¼0
P
F
(b)
r ¼ 1: ni¼0 F k;2iþ1 ¼ k;2nþ2
k
For the classical Fibonacci sequence, k ¼ 1, it is
n
X
F 2iþ1 ¼ F 2nþ2 :
i¼0
(2) If p ¼ 2 ! a ¼ 4, then
n
X
F k;4iþr ¼
F k;4ðnþ1Þþr F k;4nþr F k;r ð1Þr F k;4r
4
k þ 4k
i¼0
(a)
(b)
(c)
(d)
r ¼ 0:
Pn
r ¼ 1:
Pn
¼
r ¼ 2:
Pn
¼
r ¼ 3:
Pn
¼
i¼0 F k;4i
¼
i¼0 F k;4iþ1
i¼0 F k;4iþ2
i¼0 F k;4iþ3
2
F k;4nþ4 F k;4n k3 2k
k4 þ4k2
F k;4nþ5 F k;4nþ1 þk2
k4 þ4k2
F k;4nþ6 F k;4nþ2 2k
k4 þ4k2
F k;4nþ7 F k;4nþ3 þk2
k4 þ4k2
:
ð6Þ
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
It is relatively easy to prove by induction that the sequence fSk;anþr g ¼
Sk;aðnþ1Þþr ¼ Lk;a Sk;anþr þ ð1Þaþ1 Sk;aðn1Þþr þ F k;r þ ð1Þr F k;ar .
Sk;anþr ¼
n
X
F k;aiþr ¼ F k;r þ F k;aþr þ
i¼0
n
X
Pn
i¼0 F k;aiþr
, verifies the recurrence relation
Lk;a F k;aði1Þþr ð1Þa F k;aði2Þþr ¼ F k;r þ F k;aþr þ Lk;a
i¼2
n1
X
i¼1
þF k;aiþr ð1Þa
n2
X
þF k;aiþr
i¼0
¼ F k;r þ F k;aþr þ Lk;a ðSk;aðn1Þþr F k;r Þ ð1Þa Sk;aðn2Þþr :
Now considering Sk;anþr and Sk;aðnþ1Þþr :
Sk;anþr ¼ ð1 Lk;a ÞF k;r þ F k;aþr þ Lk;a Sk;aðn1Þþr ð1Þa Sk;aðn2Þþr ;
Sk;aðnþ1Þþr ¼ ð1 Lk;a ÞF k;r þ F k;aþr þ Lk;a Sk;anþr ð1Þa Sk;aðn1Þþr
by eliminating the terms ð1 Lk;a ÞF k;r þ F k;aþr , it is deduced:
Sk;aðnþ1Þþr ¼ ð1 þ Lk;a ÞSk;anþr ðLk;a þ ð1Þa ÞSk;aðn1Þþr þ ð1Þa Sk;aðn2Þþr :
So, the characteristic polynomial of sequence fSk;anþr g is r3 ¼ ð1 þ Lk;a Þr2 ðLk;a þ ð1Þa Þr þ ð1Þa Sk;aðn2Þþr , with roots r0 ¼ 1,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lk;a þ
L2k;a 4ð1Þa
Lk;a L2k;a 4ð1Þa
,
and
r
¼
. These numbers verify r1 þ r2 ¼ Lk;a , r1 r2 ¼ ð1Þa , and r 1 r2 ¼ L2k;a 4ð1Þa .
r1 ¼
2
2
2
Then, the solution for Sk;anþr is of the form Sk;anþr ¼ C 0 þ C 1 r n1 þ C 2 rn2 . Having in mind the relations between r1 and r2 , after
some algebra is obtained
C0 ¼
ðr 1 Lk;a ÞF k;aþr þ ðLk;a 1ÞF k;r
;
Lk;a 1 ð1Þa
C1 ¼
ðr 1 Lk;a ÞF k;aþr þ ð1Þa F k;r
;
ðr1 r 2 Þðr 2 1Þ
C2 ¼ ðr 2 Lk;a ÞF k;aþr þ ð1Þa F k;r
:
ðr 1 r 2 Þðr 1 1Þ
Observe that, in the case a ¼ 1, r ¼ 0, and then the recurrence becomes Snþ1 ¼ ð1 þ kÞSn ðk 1ÞSn1 Sn2 , which, for the
classical Fibonacci (that is k ¼ 1) reports Snþ1 ¼ 2Sn Sn2 .
way that in the preceding case, we can find that
Let us now consider the alternating sequence fð1Þn F k;anþr g. In a similar
F k;r ð1Þr F k;ar x
. Moreover, the following result is given:
the generating function for this alternating sequence is g a;r ðk; xÞ ¼ 1L
xþð1Þa x2
k;a
Theorem 7. Alternating sum of the k-Fibonacci numbers of order an þ r
n
X
ð1Þn F k;aðnþ1Þþr þ ð1Þnþa F k;anþr þ ð1Þrþ1 F k;ar þ F k;r
ð1Þi F k;aiþr ¼
F k;aþ1 þ F k;a1 þ ð1Þa þ 1
i¼0
which for different values of a and r reads as
Pn
ð1Þn F k;nþ1 ð1Þn F k;n 1
ð1Þi F k;i ¼
(1)
k
Pi¼0
ð1Þn F k;2nþ2 þð1Þn F k;2n k
n
i
(2)
¼ ð1Þn F k;n F k;nþ1
i¼0 ð1Þ F k;2i ¼
k2 þ4
Pn
i
n 2
(3)
i¼0 ð1Þ F k;2iþ1 ¼ ð1Þ F k;nþ1
n
Pn
ð1Þ
F
i
k;4nþ2 k
(4)
i¼0 ð1Þ F k;4i ¼
k2 þ2
Pn
ð1Þn F k;4nþ3 þ1
i
(5)
i¼0 ð1Þ F k;4iþ1 ¼
k2 þ2
Pn
ð1Þn F k;4nþ4
i
(6)
i¼0 ð1Þ F k;4iþ2 ¼
k2 þ2
Pn
ð1Þn F k;4nþ5 þ1
i
(7)
i¼0 ð1Þ F k;4iþ3 ¼
k2 þ2
3. Conclusions
We have studied the subsequences of k-Fibonacci numbers with indexes in an arithmetic sequence. In a compact and direct 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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