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However, Li and Wang (2014) developed a new class of generalized Fibonacci sequence by considering a unimodal maps and show that, in some circumstances, the map has no absolutely continuous invariant probability measure. They show an important dynamical structure in the model that is the properties of the absolute continuous invariant measure.

The other extension of Fibonacci sequence has been done by Bruin (2007) and Bruin and Volkova (2005) who developed a Fibonacci-like map or a kneading map. On the other hand, the Fibonacci sequence also has been extended by generalizing the integer to real and complex numbers ( Harman, 1981, Asci and Gurel, 2013), quarternion ( Halici, 2012), generalized quarternion ( Akyigit et al., 2014, Tan et al., 2016). The n-nacci sequence has found application in coin tossing problem ( Weisstein, a, Weisstein, b Griffiths, 2011). For example, the Fibonacci sequence has been extended to tribonacci, tetranacci, and other higher order n-nacci sequences ( Wolfram, 1998). The extension of the Fibonacci sequence is also widespread and penetrated many branches of mathematics including dynamical system. Given that it has many applications, many studies have been conducted to extend the sequence. The ratio can also be found in modern applications such as financial analysis and plastic surgery ( Letzter, 2016). The ratio has been found in many areas of applications such as in analyzing the proportions of natural objects and man-made systems. It is the ratio of the consecutive numbers in the Fibonacci sequence which converges to 1.61803398875 ( Livio, 2002). One of the important features arising from the Fibonacci sequence is the Golden Ratio. The complete information of the sequence can be found in the On-Line Encyclopedia of Integer Sequence ( OEIS, 2017). Lucas also showed that the Fibonacci sequence appears in the shallow diagonal of the Pascal triangle and he also defines a sequence based on the Fibonacci numbers, which is currently known as Lucas number. The first mathematician who called it Fibonacci sequence is Edouard Lucas in 19-th century ( Gardner, 1996). It is often known as the Lame sequence ( Roberts, 1992) or Viranka number ( Goonatilake, 1998) as many other ancient mathematicians have used this sequence in their document. which first appeared in Liber Abaci book of Leonardo Pisano in 1202 ( Sigler, 2002). The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors.Fibonacci numbers or Fibonacci sequence is among the most popular numbers or sequence in mathematics.The following identities involving the Fibonacci numbers can be proved by induction. The most important identity regarding the Fibonacci sequence is its recursive definition. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre. (Note that this formula is valid for all integers.

One possible explanation for this fact is that the Fibonacci numbers are given explicitly by Binet's formula. , between successive terms of the sequence tend towards the limit, a constant often denoted (the Greek letter phi, also written ). Īnd Binet's Formula Main article: Binet's formula Because these preceding terms are uniquely defined by the recursion, one frequently sees the definition of the Fibonacci sequence given in the form, and for. This allows us to compute, for example, that, ,, and so on. This change in indexing does not affect the actual numbers in the sequence, but it does change which member of the sequence is referred to by the symbol and so also changes the appearance of certain identities involving the Fibonacci numbers.Īs with many linear recursions, we can run the Fibonacci sequence backwards by solving its recursion relation for the term of smallest index, in this case. Readers should be wary: some authors give the Fibonacci sequence with the initial conditions (or equivalently ).

This is the simplest nontrivial example of a linear recursion with constant coefficients. The Fibonacci sequence can be written recursively as and for.