In this paper, we
will see that some k -Fibonacci sequences
are related to the classical Fibonacci sequence of such way that we can express the terms
of a k -Fibonacci sequence in
function of some terms of the classical Fibonacci sequence. And the formulas
will apply to any sequence of a certain set
of k -Fibonacci sequences. Thus find k -Fibonacci sequences relating to other-Fibonacci sequences when σ'k is linearly dependent of .
Fibonacci and Lucas Numbers <i>k</i> -Fibonacci Numbers Pascal Triangle1. Introduction
-Fibonacci sequence was found by studying the recursive application of two geometrical trans-
formations used in the well-known four-triangle longest-edge (4TLE) partition. This sequence generalizes the classical Fibonacci sequence [1] [2] .
1.1. Definition
For any positive real number, the -Fibonacci sequence, say is defined recurrently by for with initial conditions.
From this definition, the polynomial expression of the first -Fibonacci numbers are presented in Table 1:
If, the classical Fibonacci sequence appears and if, the 2-Fibonacci se-
Table 1
. Polynomial expression of the first k-Fibonacci numbers
quence is the classical Pell sequence.
1.2. Metallic Ratios
The characteristic equation of the recurrence equation of the definition of the -Fibonacci numbers is
and its solutions are and.
As particulars cases [3] :
1) If, then is known as Golden Ratio and it is expressed as.
2) If, then is known as Silver Ratio.
3) If, it is and it is known as Bronze Ratio.
From now on, we will represent the classical Fibonacci numbers as instead of.
Binet identity takes the form [1] with.
1.3. Theorem 1
Power for is related to by mean of the formula
Proof. By induction. For, it is obvious. Let us suppose this formula is true until:. Then, and taking into account:
Obviously, the formulas found in [1] [2] can be applied to any -Fibonacci sequence. For example, the Iden- tities of Binet, Catalan, Simson, and D’Ocagne; the generating function; the limit of the ratio of two terms of the sequence, the sum of first “” terms, etc. However, we will see that some -Fibonacci sequences are related to a first -Fibonacci sequence so that we will can express the terms of a -Fibonacci sequence according to some terms of an initial -Fibonacci sequence. And the formulas will be applicable to any sequence of a given set of -Fibonacci sequences. For instance, we will express the terms of the 4-Fibonacci sequence in function of some terms of the classical Fibonacci sequence and these formulas will be applied to other -Fibo-naccise- quences, as for example if
2. <img src="htmlimages\6-7400519x\9553ca3a-5dc8-49a6-9db6-2644703a6e9f.png" width="26.7499995231628" height="37.3749995231628" />-Fibonacci Sequences Related to the <img src="htmlimages\6-7400519x\fa327ba3-efd5-49cf-b996-fd3f365b7335.png" width="26.7499995231628" height="37.3749995231628" />-Fibonacci Sequence
In this section, we try to find the relationships that can exist between the values of and the coefficients “” and “” such that.
We can write this last equation as
because.
Main problem is to solve the quadratic Diophantine equation for “’” and “” for each value of “”.
2.1. Theorem 2
The positive characteristic root generates new -Fibonacci sequences, for, Proof. From Formula (1) it is obtained.
For it is
Then, generates the -Fibonacci sequence.
In the same way, we can prove that generates the -Fibonacci sequence, gene- rates the -Fibonacci sequence, etc. Particularly, generates the sequences.
2.2. Theorem 3
For it is verified
Proof. Taking into account both Table 1 and Formula (1), Right Hand Side (RHS) of Equation (2) is
It is worthy of note that Equation (2) is similar to the relationship between the elements of the -Fibonacci sequence. Other versions of this equation will appear in this paper. Moreover, if we are looking for the characteristic roots of this equation, then we find
And will be function of with the coefficients depending of initial conditions for and.
2.3. k-Fibonacci Sequences Related to an Initial f-Fibonacci Sequence
From two previous theorems, the -Fibonacci sequences related to an initial -Fibonacci sequence have as the positive characteristic root or that is the same, the sequence of characteristic roots generates the -Fibonacci sequences related to the first -Fibonacci sequence.
The values of the parameter of these sequences are
and Equation (2) for this sequence takes the similar form.
Next we present the first few values of the parameter:
But these polynomials verify the relationship
where are expressed in Table 1.
The coefficients of these polynomials generate the triangle in Table 2:
Last column is the sum by row of the coefficients, and it is a bisection of the classical Lucas sequence and we will see again in this paper.
If is a term of this table, then. For instance, of the second
diagonal plus 27 of the row 5 is the 77 of the row 6.
All the first diagonal sequences are listed in [4] , from now on OEIS, but the unique antidiagonal sequences listed in OEIS are:
From this study, it is easy to find the values of “” mentioned at the beginning of this section, because
Sequence also verifies the recurrence law given in Equation (2):.
In this case, the triangle of coefficients is in Table 3 and the formto generate these numbers is the same as in table of. This triangle is formed by the odd rows of 2-Pascal triangle of [2] . The sequence of the last column is a bisection of the classical Fibonacci sequence.
First diagonal sequences and the antidiagonal sequences are listed in OEIS.
Finally, for the values of is enough to do and therefore, applying Formula (3) and the de- finition of the -Fibonacci numbers,.
Table 2
. Triangle of the coefficients of kn
1
1
1
2
1
3
4
3
1
5
5
11
4
1
7
14
7
29
5
1
9
27
30
9
76
6
1
11
44
77
55
11
199
In this case, the triangle of the coefficients of the expressions of is in Table 4.
Last column is the other bisection of the classical Fibonacci sequence.
The diagonal sequence indicates the number of terms in the expansion of and it is.
In this table, it is verified:
a)
b) , if, respectively.
c) The diagonal sequences are listed in OEIS.
d) The elements of diagonal sequence, for verify the relation
Then we will apply the results to the -Fibonacci sequences, for.
3. <img src="htmlimages\6-7400519x\9cc58ded-6a80-4eca-9600-b7e19cd88c10.png" width="26.7499995231628" height="37.3749995231628" />-Fibonacci Sequences Related to the Classical Fibonacci Sequence
In this section we try to find the relations that could exist between the values of “” and “” and “” in order that the positive characteristic root is.
In this case, Equation (2) takes the form.
3.1. Integer Solutions of Equation <img src="htmlimages\6-7400519x\cfb29683-a158-4b30-9c1d-2f55e7fc371b.png" width="144.375" height="42.7500009536743" />
The integer solutions of Equation are, being the classical Lucas se- quence.
Proof. Applying Binnet Identity, and taking into account, it is
Table 3
. Triangle of the coefficients of bn
1
1
1
2
1
1
2
3
1
3
1
5
4
1
5
6
1
13
5
1
7
15
10
1
34
6
1
9
28
35
15
1
89
Table 4
. Triangle of the coefficients of an
1
1
1
2
1
2
3
3
1
4
3
8
4
1
6
10
4
21
5
1
8
21
20
5
55
6
1
10
36
56
35
6
144
Consequently, the values of the parameter “” can also be expressed as.
Integer solutions of this equation are expressed in Table 5, where is the Golden Ratio.
3.2. On the Sequences<img src="htmlimages\6-7400519x\b6902eb8-7c5c-4187-8c2c-fed233d105f5.png" width="56.1250019073486" height="49.8750019073486" />, <img src="htmlimages\6-7400519x\a8e04655-b58d-4f5b-8be0-be2517c5755c.png" width="56.1250019073486" height="49.8750019073486" />, and <img src="htmlimages\6-7400519x\da27c501-0052-47aa-a8d5-549fde509a92.png" width="56.1250019073486" height="49.8750019073486" />
We will show some properties of the sequences of Table 5.
The sequence of values of “”, is the sequence of even Fibonacci num- bers, and is known as Bisection of Fibonacci sequence. Its elements, , have the property that are perfect squares and these numbers form the sequence that is the Bisection of the classical Lucas sequence. The sequence of sums of two consecutive terms of this sequence is 5 times the following sequence.
The sequence of values of “”, is the sequence of odd Fibonacci numbers, , and is also known as Bisection of Fibonacci sequence. The sequence of sums of two consecutive terms of this sequence is the preceding sequence.
The sequence of values of “”, is the sequence of odd Lucas numbers, or, that is the same, is the sum of two even consecutive Fibonacci numbers, and is known as Bisection Lucas Sequence. The sequence of sums of two consecutive terms of this sequence is 5 times the preceding sequence.
All these sequences verify the recurrence law given in Equation (2),.
As a consequence of this situation, if we represent as the sequence of values of, then, Equation (2) is the relation.
3.3. Relationships between the <img src="htmlimages\6-7400519x\76128e0b-6fe9-4209-88ee-94f1019b9161.png" width="26.7499995231628" height="35.625" />-Fibonacci Sequences If <img src="htmlimages\6-7400519x\3d33a288-a71d-432e-be59-090b01d5178f.png" width="110.500001907349" height="37.3749995231628" /> and the Classical Fibonacci Sequence
Applying Subsection 2.3 when in Equation (3), the sequence is the se- quence.
Consequently:
4. <img src="htmlimages\6-7400519x\610fcd18-30a9-49a7-a6c5-9aee59bbb80e.png" width="26.7499995231628" height="37.3749995231628" />-Fibonacci Sequences Related with the Pell Sequence
Repeating the previous process, we can solve the Diophantine equation and being.
Table 5
. Integer solutions of the Diophantine equation 5b2 – k2 = 4
kn = L2n+1
bn = F2n+1
an = F2n
σ1,n
1
1
0
4
2
1
11
5
3
29
13
8
76
34
21
The values obtained are showed in Table 6:
4.1. On These Quences<img src="htmlimages\6-7400519x\7b6a7aab-f33f-484a-bcc9-aeb976b50fca.png" width="56.1250019073486" height="43.6250019073486" />, <img src="htmlimages\6-7400519x\b49f4575-bf4e-4440-b0c7-a303be4d06c1.png" width="56.1250019073486" height="43.6250019073486" />, and<img src="htmlimages\6-7400519x\27d59e67-6d54-4544-ad20-f50697b0f71a.png" width="62.3750019073486" height="43.6250019073486" />.
We will show some properties of the sequences of Table 4.
is the sequence of even Pell numbers. Its elements have the property
that are perfect squares, being. The sequence of sums of
two consecutive terms of this sequence is the sequence.
is the sequence of odd Pell numbers. Its elements have the proper- ty that are perfect squares.
. Its elements are the Pell-Lucas numbers,. This sequence can be obtained by summing up two consecutive terms of the sequence A001542.
Much more interesting is the sequence obtained by dividing by 2:. This sequence has been studied in [5] and has been determined as the values whose square coincide with the sum of the first Pell numbers, and it is known as the Newman-Shanks-Williams Primes. It verifies the recurrence law with initial conditions and. The se- quence of sums of two consecutive terms of this sequence is 8 times. Its ele- ments verify the property are perfect squares,.
All these sequences verify the recurrence law (2),.
As in the preceding section, if we represent the sequence of values of “” as, then these terms verify the recurrence relation, being the Silver Ratio.
4.2. Relationships between the <img src="htmlimages\6-7400519x\aecf4db7-892b-47c4-b0f1-bf9f2839264c.png" width="26.7499995231628" height="35.625" />-Fibonacci Sequences for <img src="htmlimages\6-7400519x\9554c56e-f7b5-4b8a-b8f4-914a57db9c27.png" width="217.374992370605" height="37.3749995231628" /> and the Pell Sequence
Taking into account, it is easy to prove is the geometric sequence.
Consequently:
5. <img src="htmlimages\6-7400519x\4019fe63-7e77-47af-a855-b7b006561375.png" width="26.7499995231628" height="35.625" />-Fibonacci Sequences Related to the 3-Fibonacci Sequence
Repeating the previous process, we can solve the Diophantine equation being.
The values obtained are showed in Table 7.
Table 6
. Integer solutions of the Diophantine equation 8b2 – k2 = 4
kn = P2n + P2n+2
bn = P2n+1
an = P2n
σ2,n
2
1
0
14
5
2
82
29
12
478
169
70
5.1. On These Quences<img src="htmlimages\6-7400519x\da3d5b4c-0f7f-47f3-ac59-a19d68f0390d.png" width="56.1250019073486" height="49.8750019073486" />, <img src="htmlimages\6-7400519x\c78fed9f-4506-4e12-8655-55a081052734.png" width="56.1250019073486" height="49.8750019073486" />, and <img src="htmlimages\6-7400519x\c27dfc09-c1cb-42e2-947f-2f6e52f544c9.png" width="56.1250019073486" height="49.8750019073486" />
We will show some properties of the sequences of Table 7.
, is the sequence of even 3-Fibonacci numbers. Its elements have the property that are perfect squares,. The sequence of sums of two consecutive terms is 13 times the following sequence.
, is the sequence of the odd 3-Fibonacci numbers.
is the sequence of the odd 3-Lucas numbers. This sequence can also be expressed as 3 times the sequence.
All these sequences verify the recurrence law (Equation (2)),.
The sequence verify the relationship being the Bronze Ratio [3] .
5.2. Relationships between the k–Fibonacci Sequences for and the 3-Fibonacci Sequence
Taking into account, it is easy to prove is the geometric sequence.
Consequently:
6. Conclusions
There are infinite -Fibonacci sequences related to an initial -Fibonacci sequence for a fixed value of “”. Between these sequences, the following relations are verified:
1) The relationship is verified if and only if both following relations happen:
Relationship between “”, “”, and “”:
Diophantine equation:
2) Relationship between the positive characteristic root and the–Fibonacci numbers:
3) Second sequence related to the–Fibonacci sequence:
4) Two first values of “” are and
5) Two first values of “” are and
6) Recurrence law for the sequences:
Table 7
. Integer solutions of the Diophantine equation 13 b2 – k2 = 4
kn
bn = F3,2n+1
an = F3,2n
σ3,n
3
1
0
36
10
3
393
109
33
4287
1189
360
It is worthy of remark the fact the last sequence indicates the -Fibonacci sequence related to
the initial -Fibonacci sequence generated by the respective positive characteristic root,. From this sequence, we can obtain the sequence of -Fibonacci sequences related to: taking into account the positive characteristic root of this sequence is, the sequence of -Fibonacci sequences re- lated to this has as positive characteristic root, for. For instance: from the sequence of -Fibon- acci sequences related with the classical Fibonacci sequence (see Section 2), we can ob- tain the sequences of -Fibonacci sequences related to
4-Fibonacci sequence:
11-Fibonacci sequence:
29-Fibonacci sequence:.
ReferencesFALCON, S. AND PLAZA, A. (2007) ON THE FIBONACCI K-NUMBERS. CHAOS, SOLITONS FRACTALS, 32, 1615-1624.FALCON, S. AND PLAZA, A. (2007) THEK-FIBONACCI SEQUENCE AND THE PASCAL 2-TRIANGLE. CHAOS, SOLITONS FRACTALS, 33, 38-49.SPINADEL, V.W. (1999) THE FAMILY OF METALLIC MEANS. VIS MATH, 1.HTTP://MEMBERS.TRIPOD.COM/VISMATH/SLOANE, N.J.A. THE ONLINE ENCYCLOPEDIA OF INTEGER SEQUENCES. HTTP://OEIS.ORGFALCON S. , DIAZ-BARREIRO J.L. ,et al. (2006)FALCON, S. AND DIAZ-BARREIRO, J.L. SOME PROPERTIES OF SUMS INVOLVING PELL NUMBERS MISSOURI JOURNAL OF MATHEMATICAL OF SCIENCES 18, 33-40.
