Fibonacci power series
TL;DR: In this paper, a variation of this introductory material which involves the Fibonacci numbers is considered, where the student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions.
Abstract: A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.
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01 Jan 2001
TL;DR: The first 100 Lucas Numbers and their prime factorizations were given in this article, where they were shown to be a special case of the first 100 Fibonacci Numbers and Lucas Polynomials.
Abstract: Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.
1,250 citations
Additional excerpts
...Glaister did in 1995 [223]: x 1 − x − x2 = k kx2 + (k + 1)x − k = 0 x = −(k + 1) ± √ (k + 1)2 + (2k)2 2k ....
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TL;DR: It is proposed that, by analysing integration values from space syntax analysis with a Fibonacci mathematical sequence, it is possible to differentiate social importance between a majority of spaces that are neither segregated nor integrated.
Abstract: This paper argues that the evaluation of architectural spaces as numerical entities can identify seemingly random patterns of movement behaviour. Built upon analytical theories of space syntax that can produce spatially relevant numerical values, it is proposed that, by analysing integration values from space syntax analysis with a Fibonacci mathematical sequence, it is possible to differentiate social importance between a majority of spaces that are neither segregated nor integrated. Numerical analysis is conducted through a finance tool; Fibonacci retracement calculator. For the application of proposed methodology, a restaurant is chosen as a large public architectural space of social activity. The architectural plan is analysed with space syntax tools and the resulting values entered into a Fibonacci calculator; tracking results with the values of space syntax graph revealed several locations with mediocre values. The resulting spaces are re-evaluated using real-life financial data from the res...
7 citations
Cites background from "Fibonacci power series"
...In accordance with the inquiries gathered from relevant research on the topic (see Glaister 1995; Stevens 2002; Ball 2003), it is mathematically plausible to state that Fibonacci numbers do exist and that they work in predicting stock values....
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TL;DR: In this article, a new complex variable defined as precursive time able to correlate general relativity (GR) and quantum field theory (QFT) in a single principle was characterized.
Abstract: In this paper, a new complex variable defined as “precursive time” able to correlate general relativity (GR) and quantum field theory (QFT) in a single principle was characterized The thesis was elaborated according to a hypothesis coherent with the “Einstein’s General Theory of Relativity”, making use of a new mathematical-topological variety called “time-space” developed on the properties of the hypersphere and explained mathematically through the quaternion of Hurwitz-Lipschitz algebra In this publication we pay attention to the interaction between the weak nuclear force theory (EWT) and the nuclear mass of the Standard Model
3 citations
TL;DR: In this article, the authors introduced a family of double inequalities involving the generating function for the number of partitions into odd parts and the generator function for odd divisors for Fibonacci numbers.
Abstract: Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
3 citations
Cites background from "Fibonacci power series"
...We note [3] that this series is convergent for |x| < φ− 1, where φ is the golden ratio, i....
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Posted Content•
TL;DR: In this paper, the authors derived a formula for the evaluation of weighted generalized generalized Fibonacci sums of the type $S_k^n (w,r) = \sum{j = 0}^k {w^j j^r G_j{}^n }.
Abstract: We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.
1 citations
Cites background from "Fibonacci power series"
...[7] P. Glaister, Fibonacci numbers-finite and infinite series, International Journal of Mathematical Education in Science and Technology 27:3 (1996), 429–441....
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...[6] P. Glaister, Fibonacci power series, The Mathematical Gazette 79:486 (1995), 521–525....
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...Dropping the last two terms in Corollary 2.2, we have S∞(w, 1) = ∞ ∑ j=0 wjjGj = 2− w (w2 + w − 1)2w 2G0 + w2 + 1 (w2 + w − 1)2wG1 , (2.15) a result that was also obtained and whose convergence was exhaustively discussed by Glaister [6, 7] for the Fibonacci case, (G=F)....
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...15) a result that was also obtained and whose convergence was exhaustively discussed by Glaister [6, 7] for the Fibonacci case, (G=F)....
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