On the Fibonacci k-numbers
Introduction
In the present days there is a huge interest of modern science in the application of the Golden Section and Fibonacci numbers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The Fibonacci numbers Fn are the terms of the sequence {0, 1 , 1, 2 , 3, 5 , …} wherein each term is the sum of the two preceding terms, beginning with the values F0 = 0, and F1 = 1. On the other hand the ratio of two consecutive Fibonacci numbers converges to the Golden Mean, or Golden Section, , which appears in modern research in many fields from Architecture, Nature and Art [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] to physics of the high energy particles [31], [32], [33] or theoretical physics [34], [35], [36], [37], [38], [39], [40], [41].
As an example of the ubiquity of the Golden Mean in geometry we can think of the ratio between the length of a diagonal and a side of a regular pentagon. The paper presented here was originated for the astonishing presence of the Golden Section in a recursive partition of triangles in the context of the finite element method and triangular refinements.
Grid generation and, in particular, the construction of ‘quality’ grids is a major issue in both geometric modeling and engineering analysis [42], [43], [44], [45], [46]. Many of these methods employ forms of local and global triangle subdivision and seek to maintain well shaped triangles. The four-triangle longest-edge (4TLE) partition is constructed by joining the midpoint of the longest-edge to the opposite vertex and to the midpoints of the two remaining edges [47], [48]. The two subtriangles with edges coincident with the longest-edge of the parent are similar to the parent. The remaining two subtriangles form a similar pair that, in general, are not similar to the parent triangle. We refer to such new triangle shapes as ‘dissimilar’ to those preceding. The iterative partition of obtuse triangles systematically improves the triangles in the sense that the sequence of smallest angles monotonically increases, while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration [44], [48].
In this paper, we show the relation between the 4TLE partition and the Fibonacci numbers, as another example of the relation between geometry and numbers. The use of the concept of antecedent of a (normalized) triangle is used to deduce a pair of complex variable functions. These functions, in matrix form, allow us to directly and in an easy way, present many of the basic properties of some of the best known recursive integer sequences, like the Fibonacci numbers and the Pell numbers.
Section snippets
Normalized triangles, antecedents and complex valued functions
Since we were interested in the shape of the triangles, each triangle is scaled to have the longest-edge of unit length. In this form, each triangle is represented for the three vertices: (0, 0), (1, 0) and z = (x, y). Since the two first vertices are the extreme points of the longest-edge, the third vertex is located inside two bounding exterior circular arcs of unit radius, as shown in Fig. 1. In the following, for any triangle t, the edges and angles will be respectively denoted in decreasing
k-Fibonacci numbers
In this section, a new generalization of the Fibonacci numbers is introduced. It should be noted that the recurrence formula of these numbers depend on one integral parameter instead of two parameters. We shall show that these numbers are related with the complex valued functions given above, and then, in some sense, with the 4TLE partition of normalized triangles. Definition 5 For any integer number k ⩾ 1, the kth Fibonacci sequence, say {Fk,n}n∈N is defined recurrently by
Properties from the determinant of matrix (Rk−1 · L)n
For the shake of clarity we note in the sequel by T the matrix Rk−1 · L. In this section, we shall study some properties for the kth Fibonacci sequences which are directly obtained from the determinant of matrices Tn = (Rk−1 · L)n, that is, from the associated matrices to transformations fR and fL. Proposition 7 Catalan identity . Proof If in Proposition 6 n is changed by n + r, the following matrix is obtained:and
Properties by summing up matrices (Rk−1 · L)n
In this section, we shall show some properties for the sum of the terms of the kth Fibonacci sequences, obtained by summing up the first n matrices (Rk−1 · L)n. Proposition 8 . Proof Note that the term a12 in matrix Tn = (Rk−1 · L)n is precisely Fk,n. Let Sn be the sum of the first n matrices Tj = (Rk−1 · L)j. That is, Sn = T + T2 + ⋯ + Tn. The argument here is the same that used in the proof of the sum of the n first terms of a geometric numerical progression: Since SnT = T2 + T3 + ⋯ + Tn + Tn+1, then Sn(T − I2) = Tn+1 − T
Properties from the product of matrices (Rk−1 · L)n
In this section, we shall prove some interesting properties of the kth Fibonacci sequences which may be easily deduced from the product of matrices of the form (Rk−1 · L)n. The first property is called convolution product: Proposition 14 Proof Given the matrices (Rk−1 · L)n, (Rk−1 · L)m as Eq. (1), and considering the term a12 of the product (Rk−1 · L)n × (Rk−1 · L)m, which is equal to the term a12 of matrix (Rk−1 · L)n+m we get the result. □
Particular cases:
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If k = 1, for the classic Fibonacci sequence
Conclusions
New generalized kth Fibonacci sequences have been introduced and studied. Many of the properties of these sequences are proved by simple matrix algebra. This study has been motivated by the arising of two complex valued maps to represent the two antecedents in an specific four-triangle partition.
Acknowledgement
This work has been supported in part by CYCIT Project number MTM2005-08441-C02-02 from Ministerio de Educación y Ciencia of Spain.
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