Dorota Bród (Rzeszów)
On two-parameters generalization of Fibonacci numbers
Abstract In this paper we introduce a new two-parameters generalization of Fi- bonacci numbers – distance s-Fibonacci numbers Fs(k, n). We generalize the known distance Fibonacci numbers by adding an additional integer parameter s. We give combinatorial and graph interpretations of these numbers. Moreover, we present some properties of distance s-Fibonacci numbers, which generalize known proper- ties of classical Fibonacci and Padovan numbers.
2010 Mathematics Subject Classification: 11B37; 11C20; 15B36; 05C69.
Key words and phrases: Fibonacci numbers; Padovan numbers; distance Fibonacci numbers; generalized Fibonacci numbers; generating function; matrix generator.
1. Introduction The well-known Fibonacci sequence {Fn} is defined by the ecurrence relation F (n) = F (n − 1) + F (n − 2) for n 2 with initial conditions F (0) = F (1) = 1. In 1843 Binet derived a formula F (n) = αα−βn−βn, where α = 1+
√ 5
2 , β = 1−
√ 5
2 are the roots of the characteristic equation x2 − x − 1 = 0. There are many generalizations of the Fibonacci sequence in the literature with respect to one, two or more parameters, see Bravo et al.(2015),Falcon(2014),Falcon and Plaza(2009),Falcón and Plaza(2007), Horadam(1965). The authors have generalized the second order recurrence in two ways: first by preserving the recurrence relation and second by preserving the initial conditions.
Miles(1960) defined k-generalized Fibonacci numbers fn as follows fn = Pk
j=1
fn−j for n > k 2 with f0 = f1= . . . = fk−2= 0, fk−1= fk = 1.
Horadam(1965) considered the generalized Fibonacci sequence {wn} de- fined by the following recurrence w0 = a, w1 = b, wn = pwn−1+ qwn−2 for n 2, where a, b, p, q are arbitrary integers. It is easily seen that from this recurrence, apart from Fibonacci numbers, we obtain definitions of famous sequences, for example Lucas, Pell, Padovan, Perrin, Jacobsthal sequences.
Er(1984) introduced k sequences of generalized order-k Fibonacci num-
bers as follows gin=
k
P
j=1
gin−j for n > 0 and 1 ¬ i ¬ k, and for 1 − k ¬ n ¬ 0
gni =
( 1 if n = 1 − i, 0 otherwise,
where gni is the nth term of the ith sequence. Er determined the generating matrix for the sequence {gni}.Kili¸c and Ta¸sci(2006) gave the generalized Binet formulas and the combinatorial representations for the generalized order-k Fibonacci numbers. Falcón and Plaza(2007) introduced the following one- parameter generalization of Fibonacci numbers: Fk,n= kFk,n−1+ Fk,n−2 for n 2 with Fk,0= 0, Fk,1= 1.
Among one-parametr generalizations of classical Fibonacci numbers one can find generalizations in the distance sense, i.e. such generalizations of the Fibonacci numbers that for an arbitrary integer k the n-th generalized Fibonacci number is obtained by adding two previous terms: (n − k)-th and the second choosen in such a way that the obtained recurrence generalizes the classical Fibonacci numbers. We recall some of such generalizations.
Stakhov(1999) Fibonacci p-numbers Fp(n) were considered. They were defined by the following recurrence Fp(n) = Fp(n − 1) + Fp(n − p − 1) for any given integer p 1 and n p+1 with initial conditions Fp(0) = Fp(1) = . . . = Fp(p) = Fp(p + 1) = 1. These numbers and their properties have been stud- ied by some authors (see for exampleFirengiz et al.(2014),Kilic(2008),Kilic and Stakhov(2009)). Kilic(2008) gave the generalized Binet formula, combi- natorial representations and sums of the generalized Fibonacci p-numbers by using matrix methods. Kilic and Stakhov(2009) gave relationships between the generalized Fibonacci p-numbers, Fp(n), and their sums
n
P
i=1
Fp(i). They used matrix methods, too. Moreover, they determined a class of bipartite graph whose number of 1-factors (spanning subgraphs of (2n)-vertices graph G which every vertex is of degree 1) is the generalized Fibonacci p-number.
Kwaśnik and Włoch(2000) introduced generalized Fibonacci numbers F (k, n) as follows: F (k, n) = F (k, n − 1) + F (k, n − k) for n k + 1 and F (k, n) = n + 1 for n ¬ k. Recently in Bednarz et al.(2013) there were presented the distance Fibonacci numbers F d(k, n) defined by the following recurrence F d(k, n) = F d(k, n − k + 1) + F d(k, n − k) for n k and F d(k, n) = 1 for n ¬ k − 1. In Falcon(2014) the following two-parameters generalization of Fibonacci numbers was defined: for integers k 1, n 0, r 1 and n r Fk,n(r) = kFk,n−r(r) + Fk,n−2(r) with initial conditions Fk,n(r) = 1 for n = 0, 1, . . . , r − 1, except Fk,1(1) = k. Another two-parameters gener- alization of Fibonacci numbers was introduced in [3] as follows: Let k 1, n 0 and r 1 be integers. Distance (r, k)-Fibonacci numbers Fr(k, n) are defined by the recurrence relation Fr(k, n) = rFr(k, n − 2) + rk−1Fr(k, n − k)
for n k + 1 with the following initial conditions Fr(k, 0) = Fr(k, 1) = 1,
Fr(k, n) = r[n2] for n = 2, 3, . . . , k − 2 Fr(k, k − 1) = r[k−12 ]for k 3
Fr(k, k) = rk−1+ r[k2] for k 2.
Classical Fibonacci numbers F (n) have many interesting combinatorial interpretations. We recall one of them. Let X = {1, 2, . . . , n}, n 1, be the set of n integers. Let Y∗ = {Yt∗: t ∈ T } be the family of disjoint subsets of X such that subsets Yt∗ satisfy the following conditions:
a) |Yt∗| ∈ {1, 2},
b) if |Yt∗| = 2 then Yt∗ contains two consecutive integers, c) X \ S
t∈T
Yt∗ = ∅.
The family Y∗ is a decomposition of the set X of subsets having one or two elements, such that all two-elements subsets contain consecutive integers.
Then the number of all families Y∗ is equal to the number F (n).
In this paper we introduce a new two-parameters generalization of Fi- bonacci numbers, distance s-Fibonacci numbers Fs(k, n). We show their com- binatorial interpretation related to the interpretation presented earlier. More- over, we present a graph interpretation, generating function, matrix genera- tors of the sequence {Fs(k, n)}.
Let k 2, n 0 and s 1 be integers. We define the sequence {Fs(k, n)}
of distance s-Fibonacci numbers Fs(k, n) by the recurrence relation
Fs(k, n) = sk−2Fs(k, n − k + 1) + sk−1Fs(k, n − k) for n k (1) with initial conditions Fs(k, n) = 1 for n = 0, 1, . . . , k − 2, Fs(k, k − 1) = sk−2.
For s = 1 we get F1(k, n) = F d(k, n) = F d(k, n − k + 1) + F d(k, n − k).
These numbers were introduced in [1]. If s = 1 and k = 2, then {F1(2, n)}
gives the classical Fibonacci sequence {Fn}. If s = 1 and k = 3, then {F1(3, n)} is the well-known Padovan sequence {P vn}, defined by the re- currence P v(n) = P v(n − 2) + P v(n − 3) for n 3 with P v(0) = P v(1) = P v(2) = 1.
The Table 1 includes a few first words of distance s-Fibonacci numbers Fs(k, n) for special values of k and n.
Tab.1. Distance s-Fibonacci numbers Fs(k, n)
k \ n 0 1 2 3 4 5 6 7
2 1 1 s + 1 2s + 1 s2+ 3s + 1 3s2+ 4s + 1 s3+ 6s2+ 5s + 1 4s3+ 10s2+ 6s + 1
3 1 1 s s2+ s 2s2 2s3+ s2 s4+ 3s3 4s4+ s3
4 1 1 1 s2 s3+ s2 s3+ s2 s4+ s3 2s5+ s4
5 1 1 1 1 s3 s4+ s3 s4+ s3 s5+ s4
6 1 1 1 1 1 s4 s5+ s4 s5+ s4
7 1 1 1 1 1 1 s5 s6+ s5
2. Combinatorial and graph interpretation of distance s-Fibonacci numbers Let s 1, k 3, n 2 and X = {1, 2, . . . , n}. Denote by Sn a multifamily of two-elements subsets of X of the following form
Sn= {{1, 2}, . . . , {1, 2}
| {z }
s−times
, {2, 3}, . . . , {2, 3}
| {z }
s−times
, . . . , {n − 1, n}, . . . , {n − 1, n}
| {z }
s−times
}.
Let Sns be the set of subfamilies of Snsuch that a) Sk1 ∈ Sns, where
Sk1 = h{t1, t1+ 1}, {t1+ 1, t1+ 2}, . . . , {t1+ k − 2, t1+ k − 1}i, t1 = 1, k + 1, 2k + 1, . . . , pk + 1, 0 ¬ p ¬ [k+1n ]
b) Sk−11 ∈ Sns, where
Sk−11 = h{t2, t2+ 1}, {t2+ 1, t2+ 2}, . . . , {t2+ k − 3, t2+ k − 2}i, t2 = 1, k, 2k − 1, . . . , 1 + r(k − 1), 0 ¬ r ¬ [n−k+1k−1 ]
c)
[
u1∈U1
Sk2∪ [
u2∈U2
Sk−12 ∈ Sns,
where U1 = {1, 2, . . . , n − k + 1}, U2 = {1, 2, . . . , n − k + 2}, Sk2 = h{u1, u1+ 1}, {u1+ 1, u1+ 2}, . . . , {u1+ k − 2, u1+ k − 1}i, 1 ¬ u1 ¬ n−k+1, Sk−12 = h{u2, u2+ 1}, {u2+ 1, u2+ 2}, . . . , {u2+ k − 3, u2+ k − 2}i, 1 ¬ u2 ¬ n − k + 2,
and the following conditions are satisfied:
(i) u1 6= u2, (ii) Sk2∩ Sk−12 = ∅, (iii) either S
u1∈U1
Sk2 ∪ S
u2∈U2
Sk−12 = {{1, 2}, {2, 3}, . . . , {n − 1, n}} or SSk2∪ SSk−12 = {{1, 2}, {2, 3}, . . . , {n − k + 1, n − k + 2}}.
Theorem 2.1 Let k 3, n k − 1, s 1 be integers. Then
|Sns| = Fs(k, n). (2)
Proof (by induction on n). Let k 3, n k − 1, s 1 be integers. Denote by d(n) the cardinality of the set Sns. Assume that X = {1, 2, . . . , n}. Let n = k − 1. Then we have sk−2subfamilies Sns of the multifamily Snof subsets of the set X. Hence we obtain d(k − 1) = sk−2= Fs(k, k − 1).
Let n k. Assume that equality (2) is true for an arbitrary n. We will show that d(n+1) = Fs(k, n+1). Let X = {1, 2, . . . , n+1}. Denote by δ(n+1), γ(n + 1) the number of all subfamilies Sns of the multifamily Snof subsets of X such that h{1, 2}, . . . , {k − 2, k − 1}i ∈ Sns, h{1, 2}, . . . , {k − 1, k}i ∈ Sns, respectively. It is easy to observe that
δ(n + 1) + γ(n + 1) = d(n + 1).
Moreover,
δ(n + 1) = d(n + 1 − (k − 1)) = d(n + 2 − k), γ(n + 1) = d(n + 1 − k).
By the induction hypothesis and by recurrence (1) we obtain d(n + 1) = d(n + 2 − k) + d(n + 1 − k)
= sk−2Fs(k, n + 2 − k) + sk−1Fs(k, n + 1 − k) = Fs(k, n + 1),
which ends the proof.
Note that for k = 2 numbers Fs(k, n) do not have such combinatorial interpretations.
Now we will present the graph interpretation of distance s-Fibonacci num- bers. We will use the concept of k-distance H-matchings in graphs. Let G and H be any two graphs, k 1 be an integer, a k-distance H-matching M of G is a subgraph of G such that all connected components of M are isomor- phic to H and for each two components H1, H2 of M for each x ∈ V (H1), y ∈ V (H2) holds dG(x, y) k. In case of k = 1 and H = K2 we obtain the definition of matching in the classical sense. InWłoch(2013) the generaliza- tion was introduced of an H-matching of a graph G. For a given collection H = {H1, H2, . . . , Hn} of graphs by an H-matching M of G we mean a family of subgraphs of G such that each connected component of M is isomorphic to some Hi, 1 ¬ i ¬ n. Moreover, the empty set is an H-matching of G, too.
If Hi = H for all i = 1, 2, . . . , n, then we obtain the definition of H-matching.
In this paper we consider such H-matchings, where Hi, i = 1, 2, . . . , n, belong to the class of k-vertex and (k − 1)-vertex paths, k 3.
Consider a multipath Pns, where n 2, s 1, V (Pns) = {x1, x2, . . . , xn}, E(Pns) = {{x1, x2}, . . . , {x1, x2}
| {z }
s−times
, . . . , {xn−1, xn}, . . . , {xn−1, xn}
| {z }
s−times
}. In the graph terminology the number Fs(k, n) for k 2, n k − 1, s 1 is equal to the number of special {Pk, Pk−1}-matchings of the multipath Pns such that at most k − 2 vertices xn, xn−1, . . . , xn−(k−3) do not belong to {Pk, Pk−1}- matching of the graph Pns.
Denote by q(G) the number of H-matchings of a graph G.
Theorem 2.2 Assume that s 1, k 3, n k −1 are integers. Let q(Pns) be the number of {Pk, Pk−1}-matchings of the graph Pns. Then q(Pns) = Fs(k, n).
Proof Consider a multipath Pns where vertices from V (Pns) = {x1, . . . , xn} are numbered in the natural fashion. Denote by q0(n), q00(n) the number of {Pk, Pk−1}-matchings M of Pnssuch that x1, x2, . . . , xk ∈ M , x1, x2, . . . , xk−1∈ M , respectively. It is easily seen that q0(n) + q00(n) = q(Pns).
Let M be an arbitrary {Pk, Pk−1}-matching of Pns. Consider two cases:
1) {x1, x2} ∈ E(M ) and {x1, x2} ∈ E(Pk), where Pk is a subgraph of M . Then we can choose the edge {x1, x2} and every edge of Pk on s ways. In this case M = M0∪ {Pk}, where M0 is an arbitrary {Pk, Pk−1}-matching of the graph Pns\ {x1, x2, . . . , xk} which is isomorphic to the multipath Pn−ks . Hence we obtain q0(n) = sk−1q(Pn−ks ).
2) {x1, x2} ∈ E(M ) and {x1, x2} ∈ E(Pk−1), where Pk−1 is a subgraph of M .
Proving analogously as in case 1) we obtain q00(n) = sk−2q(Pn−k+1s ).
Consequently
q(Pns) = q0(n) + q00(n) = sk−1q(Pn−ks ) + sk−2q(Pn−k+1s ).
Now we will prove that
q(Pns) = sk−1Fs(k, n − k) + sk−2Fs(k, n − k + 1).
Consider the set X = {1, 2, . . . , n} and the multifamily Sn. Assume that X corresponds to the set of vertices of the multipath Pns. Let Sns be a sub- family of the multifamily Sn (see definition of the beginning of Section 2).
Then every element of the subfamily Sns corresponds to a path of order k or k − 1 in the matching M of Pns. Hence we obtain
q(Pns) = q0(n) + q00(n) = sk−1Fs(k, n − k) + sk−2Fs(k, n − k + 1) = Fs(k, n),
which ends the proof.
3. Properties of distance s-Fibonacci numbers In this section var- ious properties of numbers Fs(k, n) are studied.
Theorem 3.1 The generating function of the sequence {Fs(k, n)} has the following form
f (x) =
k−2
P
i=0
xi
1 − sk−2xk−1− sk−1xk.
Proof Let f (x) =
∞
P
n=0
Fs(k, n)xn. Then
f (x) = Fs(k, 0) + Fs(k, 1)x + Fs(k, 2)x2+ . . . + Fs(k, k − 1)xk−1+ +
∞
P
n=k
Fs(k, n)xn.
By (1) we have
f (x) = 1 + x + x2+ . . . + xk−2+ sk−2xk−1+ +
∞
P
n=k
sk−2Fs(k, n − k + 1) + sk−1Fs(k, n − k)xn
=
k−2
P
i=0
xi+ sk−2xk−1+ sk−2
∞
P
n=k
Fs(k, n − k + 1)xn+ sk−1
∞
P
n=k
Fs(k, n − k)xn
=
k−2
P
i=0
xi+ sk−2xk−1+ +sk−2xk−1
∞
P
n=0
Fs(k, n)xn− sk−2xk−1+ + sk−1xk
∞
P
n=0
Fs(k, n)xn. Thus
f (x) =
k−2
X
i=0
xi+ (sk−2xk−1+ sk−1xk)f (x).
Hence
f (x) =
k−2
P
i=0
xi
1 − sk−2xk−1− sk−1xk.
Corollary 3.2 For s = 1 and k = 2 we get the generating function f (x) =
1
1−x−x2 of the classical Fibonacci sequence.
Corollary 3.3 For s = 1 and k = 3 we get the generating function f (x) =
1+x
1−x2−x3 of the Padovan sequence.
By recurrence (1) we obtain
Theorem 3.4 For s 1, k 2 and n 2k − 2
Fs(k, n) = sk−1Fs(k, n − k) + s2k−3Fs(k, n − 2k + 1) + s2k−4Fs(k, n − 2k + 2).
Corollary 3.5 For s = 1, k = 2 and n 1 we get known identity for classical Fibonacci numbers
F (n) = 1
2(F (n + 2) − F (n − 1)) .
Corollary 3.6 For s = 1, k = 3 and n 5 we get known identity for Padovan numbers
P v(n) = P v(n − 1) + P v(n − 5).
4. Matrix generators for distance s-Fibonacci numbers In this section we present a matrix generator for distance s-Fibonacci numbers Fs(k, n).
It is well-known that for classical Fibonacci numbers the matrix generator has the form Q =
"
1 1 1 0
#
. Moreover, Qn =
"
F (n + 1) F (n) F (n) F (n − 1)
# and det Qn = (−1)n = F (n + 1)F (n − 1) − F2(n) (the Cassini formula). The matrix generator of the distance Fibonacci numbers F d(k, n) was introduced in [1]. We apply this method for distance s-Fibonacci numbers.
Let k 2 be a fixed integer, s 1 be an integer. Let Qk = [qij] be a square matrix of size k. For a fixed integer 1 ¬ i ¬ k an element qij is equal to the coefficient of Fs(k, n − i) in the recurrence formula (1). Moreover, for j 2 we have
qij =
( 1 if j = i + 1, 0 otherwise.
By this definition we have
Q2 =
"
1 1 s 0
# , Q3 =
0 1 0 s 0 1 s2 0 0
, Q4=
0 1 0 0
0 0 1 0
s2 0 0 1 s3 0 0 0
, . . . ,
Qk=
0 1 0 . . . 0 0 0 1 . . . 0 ... ... ... . .. ...
sk−2 0 0 . . . 1 sk−1 0 0 . . . 0
.
The matrix Qk we will call the generator of the distance s-Fibonacci numbers Fs(k, n). Moreover, we define the square matrix Ak of size k as the matrix of the initial conditions of the recurrence (1) of the following form
Ak=
Fs(k, 2k − 2) Fs(k, 2k − 3) . . . Fs(k, k − 1) Fs(k, 2k − 3) Fs(k, 2k − 4) . . . Fs(k, k − 2)
... ... . .. ...
Fs(k, k) Fs(k, k − 1) . . . Fs(k, 1) Fs(k, k − 1) Fs(k, k − 2) . . . Fs(k, 0)
.
Theorem 4.1 Let s 1, k 2, n 1 be integers. Then
AkQnk =
Fs(k, n + 2k − 2) Fs(k, n + 2k − 3) . . . Fs(k, n + k − 1) Fs(k, n + 2k − 3) Fs(k, n + 2k − 4) . . . Fs(k, n + k − 2)
... ... . .. ...
Fs(k, n + k) Fs(k, n + k − 1) . . . Fs(k, n + 1) Fs(k, n + k − 1) Fs(k, n + k − 2) . . . Fs(k, n)
. (3)
Proof Let k 2 be a fixed integer. If n = 1, then by simple calculations we have
AkQk=
sk−2Fs(k, k) + sk−1Fs(k, k − 1) Fs(k, 2k − 2) . . . Fs(k, k) sk−2Fs(k, k − 1) + sk−1Fs(k, k − 2) Fs(k, 2k − 3) . . . Fs(k, k − 1)
... ... . .. ...
sk−2Fs(k, 2) + sk−1Fs(k, 1) Fs(k, k) . . . Fs(k, 2) sk−2Fs(k, 1) + sk−1Fs(k, 0) Fs(k, k − 1) . . . Fs(k, 1).
.
By (1), we obtain
AkQk=
Fs(k, 2k − 1) Fs(k, 2k − 2) . . . Fs(k, k) Fs(k, 2k − 2) Fs(k, 2k − 3) . . . Fs(k, k − 1)
... ... . .. ...
Fs(k, k + 1) Fs(k, k) . . . Fs(k, 2) Fs(k, k) Fs(k, k − 1) . . . Fs(k, 1)
.
Assume now that formula (3) is true for an arbitrary n 1 We will prove it for n + 1. By induction hypothesis we get
AkQn+1k = (AkQnk)Qk=
Fs(k, n + 2k − 2) Fs(k, n + 2k − 3) . . . Fs(k, n + k − 1) Fs(k, n + 2k − 3) Fs(k, n + 2k − 4) . . . Fs(k, n + k − 2)
... ... . .. ...
Fs(k, n + k) Fs(k, n + k − 1) . . . Fs(k, n + 1) Fs(k, n + k − 1) Fs(k, n + k − 2) . . . Fs(k, n)
· Qk.
Hence by simple calculations we obtain AkQn+1k =
sk−2Fs(k, n + k − 2) + sk−1Fs(k, n + k − 1) . . . Fs(k, n + k) sk−2Fs(k, n + k − 3) + sk−1Fs(k, n + k − 2) . . . Fs(k, n + k − 1)
... ... ...
sk−2Fs(k, n) + sk−1Fs(k, n + 1) . . . Fs(k, n + 2) sk−2Fs(k, n) + sk−1Fs(k, n + 1) . . . Fs(k, n + 1).
.
Using formula (1), we get
AkQn+1k =
Fs(k, n + 2k − 1) Fs(k, n + 2k − 2) . . . Fs(k, n + k) Fs(k, n + 2k − 2) Fs(k, n + 2k − 3) . . . Fs(k, n + k − 1)
... ... . .. ...
Fs(k, n + k + 1) Fs(k, n + k) . . . Fs(k, n + 2) Fs(k, n + k) Fs(k, n + k − 1) . . . Fs(k, n + 1)
,
which ends the proof.
Theorem 4.2 Let s 1, k 2, n 1 be integers. Then det Qk= (−1)k−1sk−1.
Corollary 4.3 For s = 1 and k = 2 we get the Cassini formula det Qn = (−1)n.
5. Acknowledgements The author wishes to thank the referees for their thorough review and very useful suggestions.
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O dwuparametrowym uogólnieniu liczb Fibonacciego.
Streszczenie W artykule wprowadzono dwuparametrowe uogólnienie klasycznych liczb Fibonacciego - odległościowe s-liczby Fibonaciego. Przedstawiono kombinato- ryczne i grafowe interpretacje tych liczb. Pokazane zostały także pewne ich własności, które uogólniają znane własności liczb Fibonacciego i liczb Padovana. Wyznaczona została także funkcja tworząca rozważanego ciągu.
2010 Klasyfikacja tematyczna AMS (2010): 11B37; 11C20; 15B36; 05C69.
Słowa kluczowe: Liczby Fibonacciego • uogólnione liczby Fibonacciego • odle- głościowe liczby Fibonacciego • liczby Padovana• funkcja tworząca• generator macierzowy.
Dorota Bród is the doctor of mathematical science at the Rzeszow University of Technology (the Faculty of Math- ematics and Applied Physics). She obtained her PhD in 2006 at the AGH University of Science and Technology of Cracow in the area of discrete mathematics. Her main re- search areas are independence and domination in graphs, extremal graphs, colourings of graphs, number theory, numbers of the Fibonacci type. References to her research
papers are found in MathSciNet underID: 676988. She is the author of two books for students (Repetytorium z analizy matematycznej, Matematyka dla studentów kierunków: architektura i urbanistyka i ochrona środowiska).
Dorota Bród
Rzeszów University of Technology
Faculty of Mathematics and Applied Physics
al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland E-mail: dorotab@prz.edu.pl
Communicated by: Jerzy Jaworski
(Received: 30th of January 2017; revised: 27th of July 2017)
