Topic

# Fibonacci number

About: Fibonacci number is a(n) research topic. Over the lifetime, 6711 publication(s) have been published within this topic receiving 65977 citation(s).

##### Papers published on a yearly basis

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01 Jan 2001

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.

1,150 citations

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Abstract: A topological index Z is proposed for a connected graph G representing the carbon skeleton of a saturated hydrocarbon. The integer Z is the sum of a set of the numbers p(G,k), which is the number of ways in which such k bonds are so chosen from G that no two of them are connected. For chain molecules Z is closely related to the characteristic polynomial derived from the topological matrix. It is found that Z is correlated well with the topological nature of the carbon skeleton, i.e., the mode of branching and ring closure. Some interesting relations are found, such as a graphical representation of the Fibonacci numbers and a composition principle for counting Z. Correlation of Z with boiling points of saturated hydrocarbons is pointed out.

1,101 citations

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01 Jan 1976

Abstract: 1. Some Preliminary Considerations. 2. Divisibility Theory in the Integers. 3. Primes and Their Distribution. 4. The Theory of Congruences. 5. Fermat's Theorem. 6. Number-Theoretic Functions. 7. Euler's Generalization of Fermat's Theorem. 8. Primitive Roots and Indices. 9. The Quadratic Reciprocity Law. 10. Perfect Numbers. 11. The Fermat Conjecture. 12. Representation of Integers as Sums of Squares. 13. Fibonacci Numbers. 14. Continued Fractions. 15. Some Twentieth-Century Developments.

901 citations

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TL;DR: The first realization of a quasiperiodic (incommensurate) superlattice is reported, which consists of alternating layers of GaAs and AlAs to form a Fibonacci sequence in which the ratio of incommensurate periods is equal to the golden mean.

Abstract: We report the first realization of a quasiperiodic (incommensurate) superlattice. The sample, grown by molecular-beam epitaxy, consists of alternating layers of GaAs and AlAs to form a Fibonacci sequence in which the ratio of incommensurate periods is equal to the golden mean $\ensuremath{\tau}$. X-ray and Raman scattering measurements are presented that reveal some of the unique properties of these novel structures.

575 citations

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TL;DR: An experiment to probe the (quasi)localization of the photon is proposed, in which optical layers are constructed following the Fibonacci sequence, which has scaling with respect to the number of layers, as well as an interesting fluctuation.

Abstract: An experiment to probe the (quasi)localization of the photon is proposed, in which optical layers are constructed following the Fibonacci sequence. The transmission coefficient has a rich structure as a function of the wavelength of light and, in fact, is multifractal. For particular wavelengths for which the resonance condition is satisfied, the light propagation has scaling with respect to the number of layers, as well as an interesting fluctuation.

444 citations