Showing papers on "Fibonacci number published in 1982"

Packing efficiency in sunflower heads

Abstract: In the mathematical description given by Vogel [1], the seeds on a sunflower head are centered at points on a “cyclotron spiral,” with constant divergence angle between any two successive seeds. It is shown that the divergence angle occurring in nature (and leading to the appearance of the Fibonacci numbers) gives the theoretically most efficient packing of the seeds. The inadequacy of Vogel's explanation for the occurrence of the Fibonacci angle is also discussed.

Waiting for the K-th consecutive success and the Fibonacci sequence of order K

Abstract: 1. Shiro Ando. "A Note on the Polygonal Numbers." The Fibonacci Quarterly 19, no. 2 (1981):180-83. 2. R. T. Hansen. "Arithmetic of Pentagonal Numbers." The Fibonacci Quarterly 8, no. 1 (1970):83-87. 3. Wm. J. LeVeque. Topics in Number Theory. Vol. I. Reading, Mass.: AddisonWesley, 1956. 4. William J. 0Donnell. "Two Theorems Concerning Hexagonal Numbers." The Fibonacci Quarterly 17, no. 1 (1979)ill'—79. 5. William J. 0Donnell. "A Theorem Concerning Octagonal Numbers." J. Recreational Math. 12, no. 4 (1979/80):271-72, 6. W. Sierpinski. "Un theoreme sur les nombres triangulaire." Elemente der Mathematik 23 (1968):31-32.'

A note on the Fibonacci quotient $Fsb{p-varepsilon }/p$

Abstract: In this note a formula analogous to Eisenstein's well known formula is presented for F p-e/p, where F n is the nth Fibonacci number (F 0 = 0, F 1 = 1), p an odd prime, and This formula is:

Deriving very efficient algorithms for evaluating linear recurrence relations using the program transformation technique

Abstract: Using the program transformation technique we derive some algorithms for evaluating linear recurrence relations in logarithmic time. The particular case of the Fibonacci function is first considered and a comparison with the conventional matrix exponentiation algorithm is made. This comparison allows us also to contrast the transformation technique and the stepwise refinement technique underlining some interesting features of the former one. Through the examples given we also explain why those features are interesting for a useful and reliable program construction methodology.

Sequences and Series

Abstract: Sequences and series have fascinated people for thousands of years. They are arrows pointing at the unreachable infinite. Aristotle described the paradoxes due to Zeno, of Achilles racing the tortoise and of “dichotomy,” both of which are answerable today as questions about infinite series. And Archimedes understood that the geometric series \(1 + {1 \over 4} + {1 \over {{4^2}}} + {1 \over {{4^3}}} + ...\) was the number 4/3. But there was very little more than that known, in theory or practice, to guide Isaac Newton when he went to work on the calculus. He used series in wholely new ways, applying his techniques of integration and differentiation to them term by term.

Efficient enhancement/lowering of integrator and differentiator time constants with Fibonacci numbers helping their evaluation

Abstract: A simple method of efficiently extending or lowering the time constants of a single operational amplifier (OA) RC ideal integrator is presented There exist two schemes in this method both of which require cascading of R -ladders to a basic integrator block It is also shown that a consecutive pair of Fibonacci numbers can be effectively used to evaluate the time constant and the gain parameter through a simple rule considerably simplifying the design procedure Finally, a complementary circuit principle has been used to obtain the differentiators to have similar extention and lowering in time constants

Waiting times and generalized Fibonacci sequences

Abstract: Suppose we have a multinormal population with k possible outcomes E/sub 1/, E/sub 2/, ..., E/sub k/ and associated probabilities ..pi../sub 1/, ..pi../sub 2/, ..., ..pi../sub k/. At each of the independent trials, one of the outcomes is observed. One may be interested in the waiting time for the occurrence of a specified event, which consists of a succession of outcomes. In this paper, we consider the probability distribution of the waiting times associated with specified events, and show how they generalize the Fibonacci, Tribonacci, ..., sequences in different ways. This is possible, since the probability generating functions of the associated waiting time random variables can be utilized to derive the probability distributions.

The orders of the Fibonacci groups

Abstract: The Fibonacci groups F(r, n) have been studied by various authors, chiefly in order to determine which ones are finite. This article contains a summary of the known results about this problem, followed by some further results obtained by the author. In particular, the orders of the groups F(r, 3) for r ≡ 2 (mod 3) and F(r, 4) for r ≡ 2 (mod 4) are determined, and various other Fibonacci groups are proved infinite by methods similar to those of Chalk and Johnson.

The number of tournaments with a unique spanning cycle

Abstract: The number of tournaments Tn on n nodes with a unique spanning cycle is the (2n-6)th Fibonacci number when n ≥ 4. Another proof of this result is given based on a recursive construction of these tournaments.

A Combinatorial Problem Concerning Processor Interconnection Networks

Abstract: The augmented data manipulator (ADM) [5] has been proposed as an interconnection scheme for microprocessors. In this note we show how the number of permutations achieved by the last stage of an n-node ADM may be derived from a Fibonacci series. This result was used in [1] to analyze the total number of permutations achieved by an entire ADM.

Search for an extremum of unimodal function of one variable in an unbounded set

Abstract: THE MINIMIZATION of a unimodal function of a single variable, specified in an unbounded set, is considered. The efficiencies of algorithm giving localization of the minimum in the intial unbounded set are estimated numerically, and the optimal algorithm is found. The algorithm makes essential use of the standard method of Fibonacci numbers and can be regarded as a generalization of the latter. A similar treatment is given of the having method as applied to localization of the point where a one-dimensional function changes sign, when this point is known to be unique but the interval in which it lies is unknown.