Showing papers on "Fibonacci number published in 1983"
TL;DR: In this paper, the authors made a numerical study of complex analytic maps by iterating the map, and found that the largest curve is a fractal, and the scaling parameter differs from those found for other types of maps.
Abstract: According to the theory of Schroder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ n th iterates of the map, whereQ n is thenth Fibonacci number.
54 citations
18 Jul 1983
TL;DR: Recursive, algebraic and arithmetic strategies for winning generalized Wythoff games in misere play are given and the notion of cedar trees, a subset of binary trees, is introduced and used for consolidating these and the normal play strategies.
Abstract: Recursive, algebraic and arithmetic strategies for winning generalized Wythoff games in misere play are given. The notion of cedar trees, a subset of binary trees, is introduced and used for consolidating these and the normal play strategies. A connection to generalized Fibonacci searches is indicated.
20 citations
TL;DR: In this paper, the Fibonacci polynomials of order k are introduced and two expansions of them are obtained, in terms of the multinomial and binomial coefficients, respectively.
Abstract: The Fibonacci polynomials of order k are introduced and two expansions of them are obtained, in terms of the multinomial and binomial coefficients, respectively. A relation between them and probability is also established. The present work generalizes results of [2] - [4] and [5].
12 citations
01 Jan 1983
TL;DR: In this article, it was shown that all solutions of (#) can be effectively determined; that is, there is an effectively computable bound B such that all solution of (*) have
Abstract: where Fm denotes the 77?th Fibonacci number, and o > 1. Without loss of generality , we may require that t be prime. The unique solution for t 2, namely (m, c) = (12, 12)5 was given by J. H. E. Cohn [2], and by 0. Wyler [11]. The unique solution for £ = 3, namely (m9 o) = (6, 2), was given by H. London and R. Finkelstein [5] and by J. C. Lagarias and D. P. Weisser [4]. A. Petho [6] showed that (#) has only finitely many solutions with t > 1, where mscs t all vary. In fact, he shows that all solutions of (#) can be effectively determined; that is, there is an effectively computable bound B such that all solutions of (*) have
11 citations
TL;DR: A simple algorithm for computing the sums of order-k Fibonacci numbers, as well as the Fib onacci numbers themselves, in log time is presented.
11 citations
10 citations
01 Jan 1983
9 citations
TL;DR: In this article, the authors give some comments on several papers dealing with ordered partitions and turn then to ordered Fibonacci partitions of {1, ߪ, n}: if d is a fixed integer, the sets A appearing in the partition have to fulfill i, j ∈ A, i ≠ j ⟹ |i-j| ≥ d.
Abstract: Ordered partitions are enumerated by Fn = Σk k !S(n, k) where S(n, k) is the Stirling number of the second kind. We give some comments on several papers dealing with ordered partitions and turn then to ordered Fibonacci partitions of {1, ߪ, n}: If d is a fixed integer, the sets A appearing in the partition have to fulfill i, j ∈ A, i ≠ j ⟹ |i-j| ≥ d. The number of ordered Fibonacci partitions is determined.
9 citations
TL;DR: The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7): given a natural number n, let 0 be the automorphism of the free group F = (xu..., xn [> of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2,...,«), given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under 9.
Abstract: The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let 0 be the automorphism of the free group F = (xu ..., xn [> of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, ... ,«). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under 9. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ^ 2, k ^ 1,
8 citations
TL;DR: New factorizations of Fibonacci numbers, Lucas numbers, and numbers of the form 2" + I are presented in this article together with the strategy (a combination of known factorization methods) used to obtain them.
Abstract: New factorizations of Fibonacci numbers, Lucas numbers, and numbers of the form 2" + I are presented together with the strategy (a combination of known factorization methods) used to obtain them.
01 Jan 1983
TL;DR: In this article, it was shown that a simple substitution automatically associates some Fibonacci identities to a class of hyperbolic ones and conversely, it is well known that by using the substitutions cos X = cosh x9 sin X = -i sinh x9 where i v-1, trigonometric identities give rise to hyperbola ones.
Abstract: It is well known that by using the substitutions cos X = cosh x9 sin X = -i sinh x9 where i v-1, trigonometric identities give rise to hyperbolic ones and conversely. This results from Euler's formulas cos X = cosh -LX and sin X = -£ sinh iX* For instance, we have the relations cosJ + sinX = 19 coshx sinh^ = 1 and sin 2J = 2 sin X cos X9 sinh 2x = 2 sinh x cosh x. Also, we shall see that a simple substitution automatically associates some Fibonacci identities to a class of hyperbolic ones. This note is more original in its form than in its conclusions. Similar methods have been used by Lucas [1], Amson [2], and Hoggatt & Bicknell [3].
TL;DR: A corollary of a general theorem proved by J. Dufresnoy and C. L. Siegel as discussed by the authors shows that Θ(tfrac{1}{2}(sqrt 5 + 1) is the smallest limit point of PV-numbers.
Abstract: A corollary of a general theorem proved by J. Dufresnoy and C. Pisot shows that\(\tfrac{1}{2}(\sqrt 5 + 1)\) is the smallest limit point of PV-numbers. A new proof of this fact is given. It is based on the, method introduced by C. L. Siegel to determine the smallest two PV-numbers and shows a close relation between the small PV-numbers and the Fibonacci series.
12 Sep 1983
TL;DR: The main thesis of as discussed by the authors is that the analysis of concurrent programs (i.e. their specifications, their proofs of correctness, deadlockfreeness, etc.) can be done by using methods very similar to those used for analysing sequential programs.
Abstract: The main thesis of this paper is that the analysis of concurrent programs (i.e. their specifications, their proofs of correctness, deadlockfreeness, etc.), can be done by using methods very similar to those used for analysing sequential programs.