A Method to Construct Generalized Fibonacci Sequences

The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an real matrix are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence to a matrix .

is an interesting numerical sequence that occurs quite frequently in many parts of nature. This sequence has a special feature; every element of this sequence, starting from the third, is the sum of its two predecessors and can be generated recursively by the formula It is clear that we need the first two terms 0 = 0, 1 = 1 and the recursive formula to define the sequence.
If we want to know the term without constructing the previous terms, we can use the unexplainable formula (see [1]): What do the irrational numbers √ 5 have to do with the original sequence?
The so-called Golden ratio = (1 + √ 5)/2 appears in nature very frequently. It is also considered the most esthetic ratio between the basis and height of a rectangle: If we replace the recursive formula by we obtain a new sequence 0, 1, 1/2, 3/4, 5/8, 11/16, 21/32, . . . and this sequence is no longer divergent; in fact, it converges to 2/3. To define a Generalized Fibonacci Sequence, we fix a natural number and two elements ( 0 , 1 , . . . , −1 ) , in the Euclidean space R . The recursive formula is 2

Journal of Applied Mathematics
The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an × real matrix are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence , , 2 , 3 , . . . to a matrix : we say → if for every ordered pair ( , ), where , ∈ {1, 2, . . . , }, the sequence of the ( , )-entries of converges to the ( , )entry of . As a particular case, we study when do we have the convergence of the powers , 2 , 3 , . . . of a Moebius transformation to a constant function.
Since | | ̸ = 0, we deduce that the vectors , , 2 , . . . , −1 are linearly independent and hence they constitute a basis for R . Calling On the other hand, consider the linear transformation : R → R defined by the formula (V) = V . Clearly, Therefore, In the next theorem, we relate and with .
Using the formula = −1 0 , we obtain any member of the corresponding Generalized Fibonacci Sequence.

Theorem 3. Consider
Journal of Applied Mathematics 3 Proof. The first row of the matrix is the following: ( 1 , 2 , . . . , ). The first column of the matrix −1 0 is where is the cofactor of the entry of in the ( , ) position.
is the entry in the (1, 1) position of the matrix . Therefore, The expression inside the square brackets coincides with To see this, develop this determinant by the first row and the last column. The coefficient of −1 is then This completes the proof.
In the particular case = 2, we obtain If we further assume that 0 = 0 and 1 = 1, we obtain = 1 In the original Fibonacci sequence, we have The roots of ( ) are 1 = 1 and 2 = −1/2. Hence, It is now clear that this last GFS converges to 2/3.
We give next a sufficient condition for the convergence of the series of a GFS.
With the help of this remark, we prove the following.
where 1 , 2 , . . . , are pairwise different real numbers. Let ( ) be the entry in the ( , ) position of the matrix . Then Be induction, the terms ( ) 1 , . . . , ( ) may be obtained using determinants of type (36). We consider the last columns of determinants (36) and we obtain . . .