Discreto Applied Mathematics 161 (2013) 2695-2701
C o n t e n t s Hsts a v a i l a b l e at S c i e n c e D i r e c t
Discrete Applied Matliematics journal homepage: vi/ww.elsevier.com/locate/dam
On a new type of distance Fibonacci numbers
CiossMark
Iwona Wtoch*, Urszula Bednarz, Dorota Brod, Andrzej Wtoch, Malgorzata Wolowiec-Musial Kzeszovv Universily ofJechnologi:. Facully of Mcnhemalics and Applied I'hysks, ai Powstaikow Warszawy 12. 35-359 Kxszow. Poland
A R T I C L E
A B S T R A C T
I N F O
Article hiaory: Received 8 January 2013 Received in revised form 19 May 2013 Accepted 27 May 20 13 Available online 17Jinie 2013
111 this paper, w e define a n e w k i n d o r F i b o i i a c c i n u m b e r s generalized in the distance sense. This g e n e r a l i z a t i o n is related to distance Fibonacci n u m b e r s and distance Lucas numbers, i n t r o d u c e d q u i t e recently. W e also s t u d y d i s t i n c t p r o p e r t i e s o f these n u m b e r s for negative integers. T h e i r representations and i n t e r p r e t a t i o n s in gi'aphs are also studied. © 2 0 1 3 Elsevier B.V. All rights reserved.
Kej'vvo/'cJ.s": Generalized Fibonacci numbers Cassini tbriiuila Pan i( ion Malchings
1. Introduction Fibonacci numbers arc given by the recurrence relation F„ = F,,.., -I F,|_.2 for n > 2 w i t h initial terms Fo = 0 and F| = 1. There are many diffei-ent generalizations in the theory of Fibonacci numbers, a m o n g other generalizations in the distance sense. In [6], Kwas'nik and W i o c h introduced generalized Fibonacci numbers F(/v. ;i) defined recursively as follows: F(k, n) = F{k. n ~ ]} + F(k. n - k) for n > k + ^ and F(k. n) = n + ^ for n < k. These numbers also have m a n y interpretations in graphs; see | 9 - n |.The graph i n t e r p r e t a t i o n is closely related to the concept o f ^-independent sets in graphs; see | 6 | . Graph interpretations of the Fibonacci numbers give new tools for s t u d y i n g properties o f numbers o f the Fibonacci type. Quite recently, another type of distance generalization of Fibonacci numbers was i n t r o d u c e d and studied in 11 ]. Distance Fibonacci numbers are denoted by Fdik. n) and defined recursively as follows: Fd{k. n) = Fd(k. n-k+]) + Fd(k, n - k) for n > k > 2, w i t h initial conditions Fd(k. Ji) = 1 for 0 < n < k - 1. Many interesting properties of these numbers were given in [ 1 ], Moreover, t w o kinds of the cyclic version of distance Fibonacci numbers Fd(k. n) w e r e introduced in [2], namely distance Lucas numbers L"'(7 1. n > 0 be integers. By (2. k)-distancc Fibonacci numbers Fjik. n). we mean generalized Fibonacci numbers defined recursively by the f o l l o w i n g relation: F2{k. n) = F2(k. n - 2J -F Fiik,
n - k)
for n > k.
w i t h initial conditions F^l/c, i) = 1 for i = 0, 1,
/< - 1.
Correspondingaulhor. Tel.:+48 178651651. t'-i?iai( addressee:
[email protected] (I. Wlocli). ubednarziS'prz.edu.pl (U. Bednarz).
[email protected] (D. Brod}, av\
[email protected],pl (A. Wloch). wolo-vviec@>pi-z.edu.pl (M. VVolowiec-Musial). 0166-218X,/.$ - see front m a t t e r ® 2013 FIsevier B.V. All rights reserved. hitp:/,'dx.doi.org/10.1016/j.dam.2013.05.029
(1)
2696
/. Wloch e[ al. / Discrete Applied Mathematics 161
(2013)2695-2701
Table 1 (2, / 3, n > 2, be integers. Then the number of all decompositions toFjik, n).
'
2697
Y( = 0, then w e have
with the rest at most one of the set X is equal
Proof. Let /< > 3, n > 2 be integers, and l e t X = { 1 , 2 , n ) . Denote by quin) the n u m b e r o f all decompositions w i t h the rest at most one of the s e t X . I f n = 2, t h e n there is the only one d e c o m p o s i t i o n o f the s e t X of the f o r m { { 1 , 2 } ) . Thus we obtain q,((2) = 1 = F2(/ 3, and suppose t h a t the equality tj; 8
28
56
70
56
28
8
1
Fig. 3. Staircase interpretation of the ¥2(2. n), FjiA.
1
n), (^2(6. n), and F2(7, n) numbers. n-2f-l
k~2
correspond to the c o m p o s i t i o n w i t h the rest at most one o f n. Hence w e have 2 ' - J2i=o
1-1
such
t-tuples. By simple calculation, w e obtain that
F2(/ 2 be integer.
det/4k =
( - 1 ) - ^ .
Then
Then F2(k, n + k) F2(k, n + k - 1 )
F2(k, n + k - 1 ) ' F2(k, n + k - 2 )
f2('
