Products Of Fibonacci Numbers Research Articles

Assembly Theory of Binary Messages

Using assembly theory, we investigate the assembly pathways of binary strings (bitstrings) of length N formed by joining bits present in the assembly pool and the bitstrings that entered the pool as a result of previous joining operations. We show that the bitstring assembly index is bounded from below by the shortest addition chain for N, and we conjecture about the form of the upper bound. We define the degree of causation for the minimum assembly index and show that, for certain N values, it has regularities that can be used to determine the length of the shortest addition chain for N. We show that a bitstring with the smallest assembly index for N can be assembled via a binary program of a length equal to this index if the length of this bitstring is expressible as a product of Fibonacci numbers. Knowing that the problem of determining the assembly index is at least NP-complete, we conjecture that this problem is NP-complete, while the problem of creating the bitstring so that it would have a predetermined largest assembly index is NP-hard. The proof of this conjecture would imply P ≠ NP since every computable problem and every computable solution can be encoded as a finite bitstring. The lower bound on the bitstring assembly index implies a creative path and an optimization path of the evolution of information, where only the latter is available to Turing machines (artificial intelligence). Furthermore, the upper bound hints at the role of dissipative structures and collective, in particular human, intelligence in this evolution.

Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers

Let {f_{k} } _{k=1}^{infty} be a Fibonacci sequence with f_{1}=f_{2}=1. In this paper, we find a simple form g_{n} such that limn→∞{(∑k=n∞ak)−1−gn}=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\lim_{n\\rightarrow\\infty} \\Biggl\\{ \\Biggl(\\sum^{\\infty}_{k=n}{a_{k}} \\Biggr)^{-1}-g_{n} \\Biggr\\} =0, $$\\end{document} where a_{k}=frac{1}{f_{k}^{2}}, frac{1}{f_{k}f_{k+m}}, or frac{1}{f_{3k}^{2}}. For example, we show that limn→∞{(∑k=n∞1f3k2)−1−(f3n2−f3n−32+49(−1)n)}=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\lim_{n\\rightarrow\\infty} \\Biggl\\{ \\Biggl(\\sum^{\\infty}_{k=n}{ \\frac {1}{f_{3k}^{2}}} \\Biggr)^{-1}- \\biggl(f_{3n}^{2}-f_{3n-3}^{2}+ \\frac {4}{9}(-1)^{n} \\biggr) \\Biggr\\} =0. $$\\end{document}

Refined estimates of the number of repetition-free boolean functions in the full binary basis {&, ∨, ⊕, −

We consider repetition-free Boolean functions in the basis {&, ∨, ⊕, −}, and prove a formula expressing the number of such functions of n variables as a product of Fibonacci numbers. These products are estimated; as a result, we obtain asymptotic estimates for the number of repetition-free Boolean functions. These estimates involve Euler numbers of second order and can be reduced by well-known methods to the form of an exponential-power series. These estimates can be used to construct the final asymptotics of the number of repetition-free Boolean functions in the full binary basis.

Products of Fibonacci numbers with indices in an interval and at most four omitted being a power

Here, we find all instances in which a product of Fibonacci numbers with indices in an interval of length k and at most four of them omitted is a perfect power.

Spanning forests and the golden ratio

For a graph G, let f ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f ij 's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F = ( f ij ) n × n / f , where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F - 1 = I + L , where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ - 1 or to 1 - φ - 1 , where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.

Diagnostic value of respiratory heart arrhythmia : G. Rabending, H. Kloeckner, G. Reichel and A. Wagner (Greifswald, GDR)

For a graph G, let fij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the fij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(fij)n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ-1 or to 1-φ-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.