![]() Taking into account both Table 1 and Formula (1), Right Hand Side (RHS) of Equation (2) is Main problem is to solve the quadratic Diophantine equation In this section, we try to find the relationships that can exist between the values ofĪnd the coefficients “ ” and “ ” such that. Fibo-naccise- quences, as for example if 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 And the formulas will be applicable to any sequence of a given set of Fibonacci sequence according to some terms of an initial Fibonacci sequence so that we will can express the terms of a Fibonacci sequences are related to a first For example, the Iden- tities 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 “ ” terms, etc. Obviously, the formulas found in can be applied to any Let us suppose this formula is true until. įrom now on, we will represent the classical Fibonacci numbers as Is known as Golden Ratio and it is expressed as. The characteristic equation of the recurrence equation of the definition of the Polynomial expression of the first k-Fibonacci numbers. Fibonacci numbers are presented in Table 1: įrom this definition, the polynomial expression of the first This sequence generalizes the classical Fibonacci sequence. Was found by studying the recursive application of two geometrical trans-įormations used in the well-known four-triangle longest-edge (4TLE) partition. And the formulas will apply to any sequence of a certain set of Fibonacci sequence in function of some terms of the classical Fibonacci sequence. Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a Received revised 23 June 2014 accepted 13 July 2014 This work is licensed under the Creative Commons Attribution International License (CC BY). So, we return the n-1 index element instead of the n index element.Department of Mathematics, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, SpainĮmail: © 2014 by author and Scientific Research Publishing Inc. Hence, when we say ’nth Fibonacci number’ or ’nth element of the array’, the element holds the index value n-1. The i-th element’s value is equal to the sum of the previous two elements in that array.Īs the loop iterates, each value of the array is filled.įinally, after the loop terminates, the value with index n-1 is returned.Īs explained before, The array index value starts from 0, but we start counting from 1. Next, in the loop, we assign the value of each element of the array. As explained in the previous code, the loop will start with the value i=2 and end with the value i=n. Now, we take a for loop with the variable i in range(2,n+1). Hence, we leave the 1st number (index 0) and change the 2nd number (index 1) to the value 1. Hence, we give an additional space because the nth Fibonacci number will have the index value n+1.Īt the creation of the array, all values are ‘0’ by default. But, the index of an array starts from 0. We start by creating an array FibArray of size n+1 – This is because, when we say nth Fibonacci number’, we start counting from 1. In this method, an array of size n is created by repeated addition using the for loop. Once the loop is terminated, the function returns the value of b, which stores the value of the nth Fibonacci number. ![]() This process continues and value 3 keeps reassigning until the loop terminates.To put it more simply, after c becomes a+b, a = b and b = c. Subsequently, the value of b is reassigned to the value of c. Once c takes the value of a+b, the value of a is reassigned to b. Consider the series to be quite literally in the sequence of a, b, & c.This variable is used to store the sum of the previous two elements in the series. Over here, we take a storage variable c.This range function means the loop starts with the value 2 and keeps iterating until the value n-1. If n is greater than 2, we take a ‘for’ loop of i in range( 2,n). Then, we return 0 for input value n=1 and 1 for input value n=2 using if-else statements.We take 2 pre-assigned variables a=0 and b=1 – they are the 1st & 2nd elements of the series.However, dynamic programming uses recursion to achieve repeated addition, while this method uses the `for loop’. This method is almost entirely the same as dynamic programming. ![]()
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