![]() ![]() , and is also known as Bisection of Fibonacci sequence. Is the sequence of odd Fibonacci numbers, The sequence of sums of two consecutive terms of this sequence is 5 times the following sequence. That is the Bisection of the classical Lucas sequence. Its elements,Īre perfect squares and these numbers form the sequence Of even Fibonacci num- bers, and is known as Bisection of Fibonacci sequence. We will show some properties of the sequences of Table 5. Integer solutions of this equation are expressed in Table 5, where Triangle of the coefficients of a n.Ĭonsequently, the values of the parameter “ ” can also be expressed as. įirst diagonal sequences and the antidiagonal sequences are listed in OEIS.Īnd therefore, applying Formula (3) and the de- finition of the The sequence of the last column is a bisection of the classical Fibonacci sequence. This triangle is formed by the odd rows of 2-Pascal triangle of. In this case, the triangle of coefficients is in Table 3 and the formto generate these numbers is the same as in table of. ![]() For instance,ĭiagonal plus 27 of the row 5 is the 77 of the row 6.Īll the first diagonal sequences are listed in, from now on OEIS, but the unique antidiagonal sequences listed in OEIS are:įrom this study, it is easy to find the values of “ ” mentioned at the beginning of this section, becauseĪlso verifies the recurrence law given in Equation (2). Last column is the sum by row of the coefficients, and it is a bisection of the classical Lucas sequence The coefficients of these polynomials generate the triangle in Table 2: Next we present the first few values of the parameter :īut these polynomials verify the relationship The values of the parameter of these sequences areĪnd Equation (2) for this sequence takes the similar form. Fibonacci sequences related to the first Or that is the same, the sequence of characteristic roots Fibonacci sequence have as the positive characteristic root Fibonacci sequences related to an initial k-Fibonacci Sequences Related to an Initial f-Fibonacci Sequence With the coefficients depending of initial conditions forĢ.3. Moreover, if we are looking for the characteristic roots of this equation, then we find Other versions of this equation will appear in this paper. It is worthy of note that Equation (2) is similar to the relationship between the elements of the 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). Department of Mathematics, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, SpainĮmail: © 2014 by author and Scientific Research Publishing Inc.
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