![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
Abstract
In this paper we study one parameter generalization of the Fibonacci numbers, Lucas numbers which generalizes the Jacobsthal numbers, Jacobsthal–Lucas numbers simultaneously. We present some their properties and interpretations also in graphs. Presented results generalize well-known results for Fibonacci numbers, Lucas numbers, Jacobsthal numbers and Jacobsthal–Lucas numbers.
1 Introduction and preliminary results
The Fibonacci numbers are defined by the recurrence relation
, for
with
,
.
The th Lucas number
is defined recursively by
for
with initial terms
,
.
Apart from the Fibonacci numbers and the Lucas numbers the well-known are the Jacobsthal numbers and the Jacobsthal–Lucas numbers.
For an integer the
th Jacobsthal number
is defined recursively by
, for
with
. The
th Jacobsthal–Lucas number
is defined by
, for
with
.
Let us consider one parameter generalization of the Fibonacci numbers.
Let ,
be integers. The
th generalized Fibonacci number
is defined recursively as follows
(1)
(1) for
with initial conditions
and
.
It is interesting to note that generalizes the Fibonacci numbers and the Jacobsthal numbers. If
then
and for
holds
. For these reasons numbers
also are named as generalized Jacobsthal numbers.
In the same way we can define the generalized Lucas numbers, which are a generalization of Lucas numbers and Jacobsthal–Lucas numbers.
Let ,
be integers. The
th generalized Lucas number
is defined recursively as follows
(2)
(2) for
with initial conditions
and
.
If then
and for
holds
.
Since the characteristic equation of relations (1) and (2) is , so roots of it are
,
. Consequently for
the direct formulas (named also as Binet formulas) for
and
have the forms
(3)
(3)
(4)
(4)
The Fibonacci numbers have many interpretations also in graph theory, see Citation[1–6]. It was shown that the generalized Fibonacci numbers are related to the Merrifield–Simmons index of a special graph product, see for details Citation[7]. In this paper we show another graph interpretation of
.
Only undirected, connected simple graphs are considered. A subset is independent if for each
, there is no edge between them. Moreover the empty set and every subset containing exactly one vertex are independent. Let
be an integer. Let us consider
copies of edgeless graph
of order
,
, denoted by
, with
for
. Then
is a graph such that
and
. Let
be the number of all independent sets
of
such that
for
.
Theorem 1
Let ,
be integers. Then
(5)
(5)
Proof
By Induction on Let
be as in the statement of the theorem. If
then the result is obvious. Suppose that
and let
for
. We shall show that
. Let
be an arbitrary independent set such that
for
. We consider the following cases.
1. |
| ||||
2. |
|
Finally which ends the proof. □
2 Identities for generalized Fibonacci and Lucas numbers
In this section we give some properties of generalized Fibonacci numbers and generalized Lucas numbers.
Theorem 2
Let ,
be integers. Then
(6)
(6)
Proof
By Induction on For
we have
.
Let be given and suppose that (6) is true for all
. We shall show that (6) holds for
. Using induction’s assumption for
and
and (1), (2) we have
Thus, (6) holds for , and the proof of the induction step is complete. □
Theorem 3
Let ,
be integers. Then
(7)
(7)
Proof
By Induction on For
we have
.
Let be given and suppose that (7) is true for all
. We shall show that (7) holds for
. Using induction’s assumption for
and
and (1), (2) we have
Thus, (7) holds for , and the proof of the induction step is complete. □
Theorem 4
Let ,
be integers. Then
(8)
(8)
Proof
Using (1) we have .
For integers we obtain
Adding these equalities we obtain (8). □
In the same way one can easily prove the next theorem.
Theorem 5
Let ,
be integers. Then
(9)
(9)
From the above theorems we obtain the well-known identities for Fibonacci numbers, Lucas numbers, Jacobsthal numbers and Jacobsthal–Lucas numbers.
Corollary 6
Let be an integer. Then
Some identities for numbers and
can be found using their matrix generators.
For integers and
let
be a matrix with entries being generalized Fibonacci numbers.
Theorem 7
Let ,
be integers. Then
Proof
By Induction on If
then the result is obvious. Assume that
We shall show that
By simple calculation using induction’s hypothesis we have
which ends the proof. □
This generator immediately gives the Cassini formula for the generalized Fibonacci numbers.
Corollary 8
Let ,
be integers. Then
If and
then we obtain the well-known Cassini formulas for the Fibonacci numbers and the Jacobsthal numbers, respectively.
Corollary 9
Let be an integer. Then
Analogously we can define the matrix generator and the Cassini formula for the generalized Lucas numbers.
For integers and
let
be a matrix with entries being generalized Lucas numbers.
Theorem 10
Let ,
be integers. Then
Corollary 11
Let ,
be integers. Then
If and
then we have the well-known Cassini formulas for the Lucas numbers and the Jacobsthal–Lucas numbers, respectively.
Corollary 12
Let be an integer. Then
References
- Dosal-TrujilloL.A.Galeana-SánchezH., The Fibonacci numbers of certain subgraphs of circulant graphs AKCE Int. J. Graphs Combin. 122015 94–103
- KwaśnikM.WłochI., The total number of generalized stable sets and kernels of graphs Ars Combin. 552000 139–146
- ProdingerH.TichyR.F., Fibonacci numbers in graphs Fibonacci Quart. 201982 16–21
- SkupieńZ., Sums of powered characteristic roots count distance-independent circular sets Discuss. Math. Graph Theory 332013 217–229
- WłochA., On generalized Fibonacci numbers and k-distance Kp-matchings in graphs Discrete Appl. Math. 1602012 1399–1405
- WłochA., Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers Appl. Math. Comput. 2192013 5564–5568
- Szynal-LianaA.WłochI., On distance Pell numbers and their connections with Fibonacci numbers Ars Combin. CXIIIA2014 65–75