Relationships between Some k-Fibonacci Sequences

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 we find ′ k -Fibonacci sequences relating to other k -Fibonacci sequences when ′ σ k is linearly dependent of 2 4 2 k k k + + = σ .


Definition
For any positive real number k , the k -Fibonacci sequence, say { } From this definition, the polynomial expression of the first k -Fibonacci numbers are presented in Table 1: , the classical Fibonacci sequence { } 0,1,1, 2, 3, 5,8, appears and if 2 k = , the 2-Fibonacci se-Table 1. Polynomial expression of the first k-Fibonacci numbers.

Metallic Ratios
The characteristic equation of the recurrence equation of the definition of the k -Fibonacci numbers is As particulars cases [3]: is known as Golden Ratio and it is expressed as Φ .
and it is known as Bronze Ratio.
From now on, we will represent the classical Fibonacci numbers as n F instead of 1,n F .

Theorem 1
Proof.By induction.For 1 n = , it is obvious.Let us suppose this formula is true until: . Then, and taking into account 2   1 0 Obviously, the formulas found in [1] [2] can be applied to any k -Fibonacci sequence.For example, the Identities 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 " n " terms, etc.However, we will see that some k -Fibonacci sequences are related to a first k -Fibonacci sequence so that we will can express the terms of a k -Fibonacci sequence according to some terms of an initial k -Fibonacci sequence.And the formulas will be applicable to any sequence of a given set of k -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 k -Fibo-naccisequences, as for example if 11, 29, 76,199, k = 

′ k -Fibonacci Sequences Related to the k -Fibonacci Sequence
In this section, we try to find the relationships that can exist between the values of k′ and the coefficients " a " Main problem is to solve the quadratic Diophantine equation ( ) and " b " for each value of " k ".

Theorem 2
The positive characteristic root Then, 3 k σ generates the ( ) In the same way, we can prove that 5 k σ generates the ( ) , , , , F F F F  .

Theorem 3
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 k -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
, 2 k n F + will be function of 2 k σ with the coefficients depending of initial conditions for 0 n = and 1 n = .

k-Fibonacci Sequences Related to an Initial f-Fibonacci Sequence
From two previous theorems, the k -Fibonacci sequences related to an initial k -Fibonacci sequence have as the positive characteristic root 2 1   n k σ + or that is the same, the sequence of characteristic roots , , ,  generates the k -Fibonacci sequences related to the first k -Fibonacci sequence.The values of the parameter of these sequences are 2) for this sequence takes the S. Falcon similar form ( ) Next we present the first few values of the parameter n k :   All the first diagonal sequences are listed in [4], from now on OEIS, but the unique antidiagonal sequences listed in OEIS are: ( ) In this case, the triangle of coefficients is in Table 3 and the formto generate these numbers is the same as in table of n k .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 { } 1,1, 2, 3, 5,8,13, 21, .First diagonal sequences and the antidiagonal sequences are listed in OEIS.
Finally, for the values of n a is enough to do 2 and therefore, applying Formula (3) and the de- In this case, the triangle of the coefficients of the expressions of n a is in Table 4.Last column is the other bisection of the classical Fibonacci sequence.
The diagonal sequence { } 1, , n  indicates the number of terms in the expansion of ( ) c) The diagonal sequences are listed in OEIS.
d) The elements of rth diagonal sequence, for  Then we will apply the results to the k -Fibonacci sequences, for 1, 2,3, 4 k = .

k -Fibonacci Sequences Related to the Classical Fibonacci Sequence
In this section we try to find the relations that could exist between the values of " k " and " a " and " b " in order that the positive characteristic root k In this case, Equation (2) takes the form    Consequently, the values of the parameter " k " can also be expressed as

Integer Solutions of Equation
Integer solutions of this equation are expressed in Table 5, where 1 1 5 2 σ + = = Φ is the Golden Ratio.We will show some properties of the sequences of Table 5.

On the
 The sequence of values of " a ", { }  , , ,

Relationships between the k -Fibonacci Sequences If
, , , ,

k -Fibonacci Sequences Related with the Pell Sequence
Repeating the previous process, we can solve the Diophantine equation We will show some properties of the sequences of Table 4. .This sequence has been studied in [5] and has been determined as the values whose square coincide with the sum of the 4

= + =
. This sequence can also be expressed as 3 times the sequence { }

Conclusions
There are infinite k -Fibonacci sequences related to an initial k -Fibonacci sequence for a fixed value of " k ". Between these sequences, the following relations are verified: 1) The relationship studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition.This sequence generalizes the classical Fibonacci sequence [1] [2].

.
For instance, 1 5 14 30+ + +of the second diagonal plus 27 of the row 5 is the 77 of the row 6.

b
From this study, it is easy to find the values of " b " mentioned at the beginning of this section, because 2 also verifies the recurrence law given in Equation (2): 11,18, 29, 47, .Proof.Applying Binnet Identity, and taking into account ( )
values obtained are showed in Table7.

3 )Table 7 .
verified if and only if both following relations happen:Relationship between " a ", " b ", and " k ": Second sequence related to the k -Fibonacci sequence: Integer solutions of the Diophantine equation 13 b 2 -k 2 = 4.

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

Table 2 .
Triangle of the coefficients of k n .

Table 3 .
Triangle of the coefficients of b n .

Table 4 .
Triangle of the coefficients of a n .

Relationships between the k -Fibonacci Sequences for
As in the preceding section, if we represent the sequence of values of " σ " as { }

Table 6 .
Integer solutions of the Diophantine equation 8b 2 -k 2 = 4.We will show some properties of the sequences of Table7.
n a , { } n b , and { } n k