Showing papers on "Fibonacci number published in 1985"
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.
590 citations
TL;DR: The Fibonacci numbers and the method for their formation were given by Virah a ṅka (between a.d. 600 and 800), Gop a la (prior to a. d. 1135) and Hemacandra (c.d., c.a.d.) as mentioned in this paper.
80 citations
TL;DR: The irregular branching pattern of the bronchial tree in multiple mammalian species is consistent with a process of morphogenetic self-similarity described by Fibonacci scaling.
Abstract: The irregular branching pattern of the bronchial tree in multiple mammalian species is consistent with a process of morphogenetic self-similarity described by Fibonacci scaling.
18 citations
TL;DR: In this paper, it was shown that if Aitken acceleration is used on the sequence (x") defined by a" = v"+1/v", the resulting sequence is a subsequence of (jc).
Abstract: Consider the sequence (t>") generated by t>"+ i = avn - bv,l_l, n 5i 2, where vi = 1, t>2 = a, with a and b real, of which the Fibonacci sequence is a special case. It is shown that if Aitken acceleration is used on the sequence (x") defined by a" = v"+1/v", the resulting sequence is a subsequence of (jc"). Second, if Newton's method and the secant method are used (with suitable starting values) to solve the equation x2 - ax + b = 0, then the sequences obtained from both of those methods are also subsequences of the original sequence.
11 citations
01 Jan 1985
8 citations
16 Dec 1985
TL;DR: Two new time and processor bounds for solving certain classes of linear recurrences for Xi's, the first n-Fibonacci numbers, can be computed in parallel in 210gn-1 units of time using only n/2 processors.
Abstract: This paper presents two new time and/or processor bounds for solving certain classes of linear recurrences. The first result provides a parallel algorithm for solving Xi=aiXi−1+di for 1≤i≤n in 210gn units of time using only 3/4n processors. The second results relate to solving Xi=Xi−1+Xi−2+di for 1≤i≤n. It is shown that Xi's can be computed in at most 310gn units of time using 5/4n processors. In the special case when di=0 for all i, it is shown that Xi's (the first n-Fibonacci numbers) can be computed in parallel in 210gn-1 units of time using only n/2 processors. These time and processor bounds compare very favourably with the previously known results for these problems.
6 citations
TL;DR: In this paper, a systematic method of the description of the temporal growth of trees is presented by defining the age of each branch as the number of bifurcation times occurring in the development process from original ancestor branch to the progeny branch.
5 citations
01 Jan 1985
TL;DR: In this paper, the authors considered a natural extension of such a sequence, in which each term is the sum of a fixed number of previous terms, randomly chosen from all previous terms.
Abstract: Although this sequence is completely deterministic, its graph resembles that of the path of a particle fluctuating randomly about the line h = n/2. Indeed, there appear to be no results on the quantitative behaviour of this sequence. Hoggatt and Bicknell [3] and Hoggatt and Bicknell-Johnson [4] studied the behavior of "r-nacci sequences, in which each term is the sum of the previous ¥> terms. A natural extension of such a sequence is one in which each term is the sum of a fixed number of previous terms, randomly chosen from all previous terms. Heyde [2] investigated martingales whose conditional expectations form Fibonacci sequences, and established almost sure convergence of ratios of consecutive terms to the golden ratio. We consider three types of sequences:
4 citations
TL;DR: In this article, a sequence of Fibonacci graphs with identical matching polynomials is defined and conditions under which they appear are given, and the appearance of such graphs may have some implications in statistical physics.
Abstract: A sequence of Fibonacci graphs is defined. A special case of Fibonacci graphs, i.e., those with identical matching polynomials, is discussed and conditions under which they appear are given. The appearance of Fibonacci graphs with identical matching polynomials may have some implications in statistical physics.
3 citations
TL;DR: A new combinatorial number, namely an associated Bell number B'(r), which enumerates the number of possible partitions of a set {1, 2,···, r} with certain constraints, is introduced, which immediately resolves the counting problem of short-circuit or bridging faults in an electrical network.
Abstract: The problem of estimating the number of all possible multiple short circuit faults in a network with a given number of lines is settled in this correspondence. A new combinatorial number, namely an associated Bell number B'(r), which enumerates the number of possible partitions of a set {1, 2,···, r} with certain constraints, is introduced. This concept immediately resolves the counting problem of short-circuit or bridging faults in an electrical network. A related combinatorial problem is also discussed which shows that under some realistic model of circuit failure, the number of possible ways the network can malfunction is closely connected to the Fibonacci sequence.
3 citations
TL;DR: In this article, a fast algorithm for computing Fibonacci numbers and their sums is presented, which runs in (3 log n) multiplicative and n additive operations for large and small n respectively.
Abstract: A fast algorithm for computing Fibonacci numbers and their sums is presented. The running time of the algorithm is (3 log n) multiplicative and n additive operations for large and small n respectively. The theoretical basis for doing so is also discussed.
TL;DR: An example of iterative dual changes in Assignment Problems which does not converge to the solution is given and properties of the Fibonacci Sequence are used.
TL;DR: In this article, a method for single-shunt fault diagnosis in an N-section recurrent resistive ladder through terminal voltage attenuation measurement and using the Fibonacci number was proposed.
Abstract: A method is proposed of single-shunt fault diagnosis in an N-section recurrent resistive ladder through terminal voltage attenuation measurement and using the Fibonacci number. The fault value is proportional to the forward and backward voltage attenuations in the ladder and some fixed Fibonacci numbers dependent on the number of sections of the ladder, thus making the evaluation very accurate. The fault location is then identified from the fault value and a single Fibonacci number, which is, in turn, related to the difference between the forward and a specified backward voltage attenuation. This unique nature of discrimination by this method, particularly of the fault section, allows a large tolerance in the attenuation measurement. Experimental results have been given along with pertinent discussions.