Generalized Fibonacci Sequences Generated from a k – Fibonacci Sequence

In this paper we will prove that all k–Fibonacci sequence contains generalized Fibonacci sequences and we will indicate the form of obtain them.

k-Lucas numbers are defined (Falcon, S. 2011) by the recurrence equation 2. On the k, l-Fibonacci Sequences Contained in the k-Fibonacci Sequences , with the initial conditions F 0 (k, l) = 0 and F 1 (k, l) = 1.If l = 1, then we obtain the k-Fibonacci sequence.
In (Falcon, S., 2009) we have proved that for any r, n ∈ N the k-Fibonacci number F k,rn is multiple of F k,n .Then, for a fixed value of r, we can find out the sequence With object to simplify the writing, we will put and the sequence generated is For instance: for k = 3 and r = 2, it is = 120, etc. and we obtain the sequence {0, 1, 11, 120, 1309, 14279, . ..}.But this sequence is a (11, −1) generalized Fibonacci sequence wich terms verify the recurrence relation In Corollary 1, we will prove that if r is an odd number, we obtain a new k-Fibonacci sequence, so and as they have been studied in (Falcon, S., 2007 (32), ( 38)).For instance, if r = 3 and k = 4, the G 4 (3) sequence is {0, 1, 76, 5777, 439128, . ..} listed in OEIS as A049669.The elements of this sequence verify the recurrence relation With the help of this Lemma, we will prove the main theorem of this paper.
Theorem 2.2 The elements of the sequence G k (r) verify the recurrence relation for n ≥ 1, with the initial conditions G k,0 (r) = 0 and G k,1 (r) = 1.
Proof: Initial conditions are obvious.
By applying the Binet's identity to the Right Hand Side (RHS) of Equation ( 2), and taking into account Corollary If r is an odd number, the sequence F k (r) is a k-Fibonacci sequence.
On the other hand, as a consequence of the Honsberger's formula, it is , being r an odd number: all subsequence obtained from the k-Fibonacci sequence when r is odd is a bisection of the corresponding k-Lucas sequence.
This circumstance does more easy to find the k-Fibonacci sequences that we can obtain of this form.For instance, for k = 2 (Pell sequence) and r = 5, it is F 2,4 = 12 and F 2,6 = 70 so we get the 82-Fibonacci sequence.

First case
Let us suppose r = 3, 5, 7, 9, . ... The second term of the generalized Fibonacci sequence obtained (n = 2), determines the order of the k-Fibonacci sequence according to the following relation proved in (Falcon, S. 2011): Finally, for k = 1, 2, 3, 4, 5 . .., we find the generalized Fibonacci sequences G k,n (r) contained in the k-Fibonacci sequence F k,n .In the Table 1, the first generalized Fibonacci sequences obtained of this form are related, where the inner number indicates the obtained k-Fibonacci sequence: It is interesting to indicate that all the row sequences of this table are bisections of the corresponding k-Lucas sequence.

Lemma
For n ≥ 2 : Proof: From the definition of the k-Lucas numbers: Then, the row sequences of this last table verify the recurrence relation With respect to the classical Fibonacci sequence, in OEIS it is indicated that 9n)  34 }, etc. where the denominators are the terms of the bisection classical Fibonacci sequence {F 2n+1 }.

Second case
Let us suppose r = 2, 4, 6, 8, 10, . ... The second term of the generalized Fibonacci subsequence obtained (n = 2), determines the order of the k-Fibonacci sequence according to the preceding formula G k,mr = F k,mr+1 + F k,mr−1 = L k,mr and taking into account F k,−1 = F k,1 = 1: Then, for k = 1, 2, 3, 4, 5 . .., we find the generalized Fibonacci subsequences G k,n (r) contained in the k-Fibonacci sequence F k,n .The following table shows the first of these sequences:

Conclusion
Here we have proved that any k-Fibonacci sequence contains infinite generalized Fibonacci sequences and we have indicated a form to obtain them.
sequences are listed in OEIS and are the even bisection of the k-Lucas sequence, L k,2n .These row sequences verify the recurrence relationG k,n+1 = (k 2 + 2)G k,n − G k,n−1 = L k,2 G k,n − G k,n−1 for n ≥ 1 with the initial conditions G k,1 = 2 = L k,0 and G k,2 = k 2 + 2 = L k,2 with k ∈ N.

Table 1 .
Generalized Fibonacci sequences for r odd

Table 2 .
Generalized Fibonacci subsequences for r even