On the Fibonacci k-numbers
TL;DR: In this paper, the authors introduced a general kth Fibonacci sequence that generalizes, between others, both the classic FIFO sequence and the Pell sequence, by studying the recursive application of two geometrical transformations used in the well-known 4TLE partition.
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 { F k , 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.
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TL;DR: In this paper, the general k-Fibonacci sequence was found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge partition.
Abstract: The general k-Fibonacci sequence { F k , n } n = 0 ∞ were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.
205 citations
Cites background from "On the Fibonacci k-numbers"
...In [40] we showed the relation between the 4-triangle longest-edge (4TLE) partition and the Fibonacci numbers, as another example of the relation between geometry and numbers....
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...These numbers extend the definition of the k-Fibonacci numbers given in [40], where k was a positive integer....
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TL;DR: In this paper, the derivatives of k-Fibonacci polynomials are presented in the form of convolution of KF-FBNs and their properties admit a straightforward proof.
Abstract: The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven. � 2007 Elsevier Ltd. All rights reserved.
122 citations
Cites background from "On the Fibonacci k-numbers"
...Function fkðxÞ 1⁄4 x 1 kx x2 is the generating function of the k-Fibonacci polynomials [20]....
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...In [20] we showed the relation between the 4-triangle longest-edge partition and the k-Fibonacci numbers, as another example of the relation between geometry and numbers....
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...Let Q be the square matrix ðRk 1 LÞ which was introduced in [20]....
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01 Jan 2011
TL;DR: In this paper, the authors examined some of the interesting properties of the k-Lucas numbers themselves as well as looking at its close relationship with k-Fibonacci numbers.
Abstract: From a special sequence of squares of k-Fibonacci numbers, the kLucas sequences are obtained in a natural form. Then, we will study the properties of the k-Lucas numbers and will prove these properties will be related with the k-Fibonaci numbers. In this paper we examine some of the interesting properties of the k-Lucas numbers themselves as well as looking at its close relationship with the k-Fibonacci numbers. The k-Lucas numbers have lots of properties, similar to those of k-Fibonacci numbers and often occur in various formulae simultaneously with latter. Mathematics Subject Classification: 11B39, 11B83
90 citations
Cites background from "On the Fibonacci k-numbers"
...}: A006190 Some of the properties that the k-Fibonacci sequences verify are summarized bellow, (see [3, 4] for details of the proofs):...
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...show some properties proven in papers [3, 4] which generalize the respective properties of the classical Fibonacci sequence....
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TL;DR: In this article, the authors introduce a matrix Q h(x ) that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers and present properties of these polynomials.
Abstract: Let h ( x ) be a polynomial with real coefficients. We introduce h ( x ) -Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h ( x ) -Fibonacci polynomials. We also introduce h ( x ) -Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ( x ) that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.
79 citations
Cites background from "On the Fibonacci k-numbers"
...Falcón and Plaza [11] introduced a general Fibonacci sequence that generalizes, among others, both the classical Fibonacci sequence and the Pell sequence....
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01 Jan 2010
TL;DR: Some new identities for k-Fibonacci numbers are obtained by Binet’s Formula and divisibility properties of these numbers have been investigated.
Abstract: In this paper, we obtain some new identities for k-Fibonacci numbers. Moreover the identities including generating functions for kFibonacci numbers have been obtained by Binet’s Formula, also divisibility properties of these numbers have been investigated. Mathematics Subject Classification: 11B39, 11B83
68 citations
Cites methods from "On the Fibonacci k-numbers"
...In [2], these general k-Fibonacci numbers {Fk,n}n=0 were found by studying the recursive application of two geometrical transformations used in the wellknown four-triangle longest-edge(4TLE) partition....
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References
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Book•
01 Jan 2002
TL;DR: Jeanneret and Ozenfant as discussed by the authors introduced a new proportional system called the Modulor, which was supposed to provide a harmonic measure to the human scale, universally applicable to architecture and mechanics.
Abstract: pursuits. He began his studies of architecture in 1905 and eventually became one of the most influential figures in modern architecture. In the winter of 1916-1917, Jeanneret moved to Paris, where he met Amedee Ozenfant, who was well connected in the Parisian haut monde of artists and intellectuals. Through Ozenfant, Jeanneret met with the Cubists and was forced to grapple with their inheritance. In particular, he absorbed an interest in proportional systems and their role in aesthetics from Juan Gris. In the autumn of 1918, Jeanneret and Ozenfant exhibited together at the Galerie Thomas. More precisely, two canvases by Jeanneret were hung alongside many more paintings by Ozenfant. They called themselves "Purists," and entitled their catalog Apres le Cubisme (After cubism). Purism invoked Piero della Francesca and the Platonic aesthetic theory in its assertion that "the work of art must not be accidental, exceptional, impressionistic, inorganic, protestatory, picturesque, but on the contrary, generalized, static, expressive of the invariant." Jeanneret did not take the name "Le Corbusier" (co-opted from ancestors on his mother's side called Lecorbesier) until he was thirty-three, well installed in Paris, and confident of his future path. It was as if he wanted basically to repress his faltering first efforts and stimulate the myth that his architectural genius bloomed suddenly into full maturity. Originally, Le Corbusier expressed rather skeptical, and even negative, views of the application of the Golden Ratio to art, warning against the "replacement of the mysticism of the sensibility by the Golden Section." In fact, a thorough analysis of Le Corbusier's architectural designs and "Purist" paintings by Roger Herz-Fischler shows that prior to 1927, Le Corbusier never used the Golden Ratio. This situation changed dramatically following the publication of Matila Ghyka's influential book Aesthetics of Proportions in Nattire and in the Arts, and his Golden Number, Pythagorean Rites and Rhythms (1931) only enhanced the mystical aspects of (4) even further. Le Corbusier's fascination with Aesthetics and with the Golden Ratio had two origins. On one hand, it was a consequence of his interest in basic forms and structures underlying Figure 79 THE GOLDEN RATIO 1 73 natural phenomena. On the other, coming from a family that encouraged musical education, Le Corbusier could appreciate the Pythagorean craving for a harmony achieved by number ratios. He wrote: "More than these thirty years past, the sap of mathematics has flown through the veins of my work, both as an architect and painter; for music is always present within me." Le Corbusier's search for a standardized proportion culminated in the introduction of a new proportional system called the "Modulor." The Modulor was supposed to provide "a harmonic measure to the human scale, universally applicable to architecture and mechanics." The latter quote is in fact no more than a rephrasing of Protagoras' famous saying from the fifth-century i.C. "Man is the measure of all things." Accordingly, in the spirit of the Vitruvian man (Figure 53) and the general philosophical commitment to discover a proportion system equivalent to that of natural creation, the Modulor was based on human proportions (Figure 79). A six-foot (about 183-centimeter) man, somewhat resembling the familiar logo of the "Michelin man," with his arm upraised (to a height of 226 cm; 7'5"), was inserted into a square (Figure 80). The ratio of the height of the man (183 cm; 6') to the height of his navel (at the midpoint of 113 cm; 3' 8.5") was taken to be precisely in a Golden Ratio. The total height (from the feet to the raised arm) was also divided in a Golden Ratio (into 140 cm and 86 cm) at the level of the wrist of a downward-hanging arm. The two ratios (113/70) and (140/86) were further subdivided into smaller dimensions according to the Fibonacci series (each number being equal to the sum of the preceding two; Figure 81). In the final version of the Modulor (Figures 79 and 81), two scales of interspiraling Fibonacci dimensions were therefore introduced (the "red and the blue series").
579 citations
"On the Fibonacci k-numbers" refers background in this paper
...(2) is known as Livio’s formula [2]....
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...• If k = 1, for the classic Fibonacci sequence is obtained: Fn+m = Fn+1Fm + FnFm 1 (Honsberger formula [2])....
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...h Particular cases: • If k = 1, for the classic Fibonacci sequence is obtained: P1 j1⁄41 F j pj 1⁄4 p p2 p 1, which for p = 10 gives: P1 j1⁄41 F j 10j 1⁄4 10 89 [2]....
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...In the present days there is a huge interest of modern science in the application of the Golden Section and Fibonacci numbers [1–19]....
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Book•
01 Jan 1989
422 citations
Book•
01 Jan 1973
TL;DR: This book combines an exposition of basic theory with a variety of applications to the physical sciences and engineering for mathematics, engineering and physical science students taking complex analysis for the first time.
Abstract: Designed for mathematics, engineering and physical science students taking complex analysis for the first time, this book combines an exposition of basic theory with a variety of applications to the physical sciences and engineering. Major changes and improvements undertaken for this new edition include: the rewriting of the proof of Cauchy's Theorem (central to complex analysis); new exercises, graded by degree of difficulty; and answers to all odd-numbered exercises now listed at the back of the book.
374 citations
"On the Fibonacci k-numbers" refers background in this paper
...Function fR is a Moebius transformation (also homography or fractional linear transformation) [49,50], while function fL is an anti-homography....
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Book•
01 Jan 1993
TL;DR: In this article, the authors present a gallery of grid generation in three dimensions and on Curves and Surfaces, with a focus on grid generation on the line and on contravariant functionals.
Abstract: Preliminaries. Application to Hosted Equations. Grid Generation on the Line. Vector Calculus and Differential Geometry. Classical Planar Grid Generation. Variational Planar Grid Generation. Tensor Analysis and Transformation Relationships. Advanced Planar Variational Grid Generation. Grid Generation in Three Dimensions. Variational Grid Generation on Curves and Surfaces. Contravariant Functionals: Alignment and Diagonalization. Tensor Coefficients. Fortran Code Directory. A Rogue's Gallery of Grids.
322 citations
TL;DR: Two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed and can be adequately combined with adaptive and/or multigrid techniques for solving finite element systems.
Abstract: Two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed. The algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used. The algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation; the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way. Proofs of the above properties are presented. The second algorithm is a simpler, improved version of the first which retains most of the properties of the latter. The algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation. In this sense they can be adequately combined with adaptive and/or multigrid techniques for solving finite element systems. Examples of the application of the algorithms are given and two possible generalizations are pointed out.
317 citations
"On the Fibonacci k-numbers" refers background in this paper
...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]....
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