Showing papers on "Fibonacci number published in 1971"
TL;DR: In this paper, a topological index Z is proposed for a connected graph G representing the carbon skeleton of a saturated hydrocarbon, where 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.
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.
1,171 citations
TL;DR: In this paper, the generalized Fibonacci numbers were studied and discussed in the context of generalized finite numbers, and the American Mathematical Monthly: Vol. 78, No. 10, pp. 1108-1109.
Abstract: (1971). On Generalized Fibonacci Numbers. The American Mathematical Monthly: Vol. 78, No. 10, pp. 1108-1109.
61 citations
01 Jan 1971
11 citations
TL;DR: A local set of rules for growing any of the many kinds of fruits in which the numbers of clockwise and counterclockwise spirals of scales, florets, or what have you, are successive Fibonacci numbers was developed by W.S. McCulloch as discussed by the authors.
Abstract: A central interest in mathematical biology is to find localized rules for growing objects having global characteristics that can be simply characterized within different reference frames. W.S. McCulloch, a few years before his death, developed (but never published) a local set of rules for growing any of the many kinds of fruits in which the numbers of clockwise and counterclockwise spirals of scales, florets, or what have you, are successive Fibonacci numbers. This article explains these rules by applying them to a five- and eight-spiral pine cone.
9 citations
5 citations
TL;DR: The interpretation of a Fibonacci tree as a stream drainage pattern leads to an exact description of all the stream order properties as mentioned in this paper. But this interpretation is not a complete one.
Abstract: The interpretation of a Fibonacci tree as a stream drainage pattern leads to an exact description of all the stream order properties by Fibonacci numbers. The asymptotic bifurcation ratio for both Horton and Strahler orders is the special constant 1 + r where r is the famous golden ratio 1.618….
5 citations
1 citations
01 Jan 1971
TL;DR: In this report, a Fibonacci coder with a binary feedback shift register is discussed and examined and the implementation of the digital data transmission systems with fibonacci Code becomes realizable.
Abstract: The author presented "A Coding Scheme for Underwater Digital Data Transmission" at the IEEE ICEOE '70 which was to propose a three valued signal transmission in the underwater environment in order to have low susceptibility against severe fluctuations of transmission speed of the transmission media. This report proposes a coding device for the above three valued code, that is, for Fibonacci Code. Though Fibonacci Code has three levels, it is possible to construct coders with binary devices because of the restrictions on the code. In this report, a Fibonacci coder with a binary feedback shift register is discussed and examined. With the proper applications of the Fibonacci coders, the implementation of the digital data transmission systems with Fibonacci Code becomes realizable.
TL;DR: In this article, the authors presented the standard formula for the general term of the Fibonacci sequence and presented a method for obtaining a formula for any linear recursive sequence in the vector space of sequences satisfying the recurrence xn+2 = Xn+I + xn.
Abstract: The work was partially supported by NSF Grant GP25664. Parts of this paper were presented to the NCTM meeting in San Diego, 1970, and to the CMC Northern Section in 1970. The Fibonacci sequence is seen as an element of a vector space of sequences satisfying the recurrence xn+2 = Xn+I + xn . This setting is used to derive the standard formula for the general term of the Fibonacci sequence and to present a method for obtaining a formula for the general term of any linear recursive sequence. Suggestions for use of this material in the classroom are included, as well as examples and exercises.
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TL;DR: This note considers, for each n = 0;1;2; : : : ; nondecreasing sequences which are rendered incomplete by the removal of any n+ 1 terms, but not by the removed of anyn terms, and calls such a sequence an n-sequence.
Abstract: In [1], it is shown that the Fibonacci sequence 1;1;2;3;5; : : : has the property that if any one term is removed from the sequence then every positive integer can be expressed as the sum of some of the terms that remain, and that if any two terms are removed, then there is a positive integer that cannot be expressed as the sum of some of the remaining terms. We describe this situation by saying that removal of any two terms from the Fibonacci sequence renders it incomplete, while removal of any one term does not. A sequence which is not incomplete will be called complete. In this note we consider, for each n = 0;1;2; : : : ; nondecreasing sequences which are rendered incomplete by the removal of any n+ 1 terms, but not by the removal of any n terms. We call such a sequence an n-sequence. Thus the Fibonacci sequence is a 1-sequence.We characterize in a simple way the set of all 1-sequences and show that there are no n-sequences for any n 2. A simple description of the set of all 0-sequences seems more difficult, and is left open.