On the Fibonacci k-numbers

https://doi.org/10.1016/j.chaos.2006.09.022Get rights and content

Abstract

We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers {Fk,n}n=0 were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.

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, τ=1+52, 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}nN is defined recurrently byFk,0=0,Fk,1=1,andFk,n+1=kFk,n+Fk,n-1

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

Fk,n+r+1Fk,n+r-1-Fk,n+r2=(-1)n+r.

Proof

If in Proposition 6 n is changed by n + r, the following matrix is obtained:(Rk-1·L)n+r=Fk,n+r+1-Fk,n+rFk,n+rFk,n+r+1-Fk,n+r-1Fk,n+r+Fk,n+r-1and |(Rk-1·L)n+r|=Fk,n+r+1Fk,n+r-1-

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

i=1nFk,i=1k(Fk,n+1+Fk,n-1).

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

Fk,n+m=Fk,n+1Fk,m+Fk,nFk,m-1.

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:

  • 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.

References (51)

  • M.S. El Naschie

    Statistical geometry of a cantor discretum and semiconductors

    Comput Math Appl

    (1995)
  • M.S. El Naschie

    Non-Euclidean spacetime structure and the two-slit experiment

    Chaos, Solitons & Fractals

    (2005)
  • M.S. El Naschie

    On the cohomology and instantons number in E-infinity Cantorian spacetime

    Chaos, Solitons & Fractals

    (2005)
  • M.S. El Naschie

    Stability analysis of the two-slit experiment with quantum particles

    Chaos, Solitons & Fractals

    (2005)
  • M.S. El Naschie

    Dead or alive: Desperately seeking Schrödinger’s cat

    Chaos, Solitons & Fractals

    (2005)
  • A. Plaza et al.

    Mesh quality improvement and other properties in the four-triangles longest-edge partition

    Comput Aided Geomet Des

    (2004)
  • V.E. Hoggat

    Fibonacci and Lucas numbers

    (1969)
  • M. Livio

    The Golden ratio: The Story of Phi, the world’s most astonishing number

    (2002)
  • A.F. Horadam

    A generalized Fibonacci sequence

    Math Mag

    (1961)
  • A.G. Shanon et al.

    Generalized Fibonacci triples

    Am Math Mon

    (1973)
  • K. Hayashu

    Fibonacci numbers and the arctangent function

    Math Mag

    (2003)
  • S. Vajda

    Fibonacci and Lucas numbers, and the Golden Section. Theory and applications

    (1989)
  • H.W. Gould

    A history of the Fibonacci Q-matrix and a higher-dimensional problem

    Fibonacci Quart

    (1981)
  • D. Kalman et al.

    The Fibonacci numbers – exposed

    Math Mag

    (2003)
  • A. Benjamin et al.

    The Fibonacci numbers – exposed more discretely

    Math Mag

    (2003)
  • Cited by (219)

    • Optimized CPU–GPU collaborative acceleration of zero-knowledge proof for confidential transactions

      2023, Journal of Systems Architecture
      Citation Excerpt :

      By considering all the performance metrics, a lookup table size of 24 KB is set in the experiment. We apply the Fibonacci [46] search algorithm for searching and updating the lookup table due to its overall good performance and fast searching speed. Fig. 14 shows that hit ratio increases as the lookup table keeps updating.

    • On k-Fibonacci graphs

      2024, Montes Taurus Journal of Pure and Applied Mathematics
    View all citing articles on Scopus
    View full text