The aim of this paper was to suggest new generalized Fibonacci sequence to a particular class of recursive sequence. Materials and Methods: The equilibrium point of the model was investigated and a new sequence.

## Using Matrices to Generate the Fibonacci Sequence

The matrix method was applied to perform the generalization. Results: The nth power of the generating matrix for this generalized Fibonacci sequence was established and some basic properties of this sequence were obtained by matrix methods. The Fibonacci sequence is a series of numbers in which a number is found by adding the two numbers before it. Initially with 0 and 1, the sequence starts from 0, 1, 1, 2, 3, 5, Knowledge of numbers is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied while growing up in North Africa.

### Fibonacci Numbers

The well-known Fibonacci sequence is a sequence of positive integers that has been studied over several years. The most and vast research field of Fibonacci numbers is defined to study the generalizations of Fibonacci numbers Bilgici 1 and Tasyurdu et al. The main aim of the present paper is to study other generalized Fibonacci sequence by matrix methods. These numbers were first studied by Horadam 3 and are called Horadam numbers.

In Silvester 4 , it has been shown that a number of the properties of the Fibonacci sequence can be derived from a matrix representation. In doing so, it showed that if:. In Demirturk 5 obtained summation formulae for the Fibonacci and Lucas sequences by matrix methods. The authors presented some important relationship between k-Jacobsthal matrix sequence and k-Jacobsthal-Lucas matrix sequence and k is the positive real number 6. In Godase and Dhakne 7 described some properties of k-Fibonacci and k- Lucas number by matrix terminology.

Borges et al. The sequence of Eq. The main aim of this paper was to study other generalized Fibonacci sequences by matrix methods.

## The Ring of Fibonacci (Fibonacci “Numbers” with Matrix Subscript) | SpringerLink

Definition: For the positive real number k, the generalized Fibonacci sequence, say S k,n defined by:. It produces two roots as:. Proof: The general form for the generalized Fibonacci sequence may be expressed in the form:. Proof: To prove this we will use Eq. Proof: Here we shall use induction on n. Indeed Eq. Now suppose that Eq.

Proof: From Eq. Proof: To prove the ongoing result we shall use introduction on n. Suppose that the result is true for n, then:. Proof: To prove the ongoing result we shall use induction on n. For this purpose we should prove the following theorem:.

Since S a square matrix and so let x be the eigen value of S and then by the Cayley Hamilton theorem on matrix, we get:. This is the characteristic equation of the generating matrix. Let a and b be the roots of the characteristic equation and also a and b are two eigen values of the square matrix S. Now we will try to find the eigen vectors corresponding to the eigen values a and b.

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First of all we calculated the eigen vectors corresponding to the eigen value of a and then:. I would suggest to comment before voting down and why? I would like to understand that algorithm, seeking help to make it clear what exactly done. Didn't downvote, but it was possibly because the logic used is more on the math side.

The code used on that page is just an implementation of the math logic.

Do these help? Blastfurnace, seen those links, Bit hard in understanding how that logic is implemented there. May be I can ask in this way, how to solve Fibonacci series in O log n time? What you need to understand is the algorithm, not the implementation.

### Fibonacci sequence that fills an n*n matrix

Featured on Meta. Unicorn Meta Zoo 9: How do we handle problem users? Linked 6. Related The solution to the equation above is:. A second order difference equation is a function of two other samples, for example. The easiest way to describe the Fibonacci sequence is with a second order difference equation.

We can solve the sequence with linear algebra. We can use eigenvectors and eigenvalues to solve the matrix power see Eigenvalues and Eigenvectors.