The problem of computing the number of sequences of various lengths that can be obtained by decimating a given binary sequence X of length n is considered. It is proven that this number is maximized iff X is an alternating sequence and that the maximum can be expressed in terms of the Fibonacci numbers. Some other upper bounds on this number are also determined, including another bound in terms...

The summation formula within Pascal triangle resulting in the Fi-bonacci sequence is extended to the q-binomial coefficients q-Gaussian triangles [1, 2]. 1 Pisa historical remark The Fibonacci sequence origin is attributed and referred to the first edition (lost) of " Fiber abaci " (1202) by Leonardo Fibonacci [Pisano] (see second edition from 1228 reproduced as Il Liber Abaci di Leonardo Pisan...

The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function of a complex variable. We experimentally study the complex dynamics of the Fibonacci numbers viewed in that manner. Attracting and repelling fixed points are related to the filled Julia set and to regions of escape time images with fascinating behavior. Introduction The Fibonacci numbers are traditionally des...

We dedicate this paper to investigate the most generalized form of Fibonacci Sequence, one of the most studied sections of the mathematical literature. One can notice that, we have discussed even a more general form of the conventional one. Although it seems the topic in the first section has already been covered before, but we present a different proof here. Later I found out that, the auxilia...

In this paper, the concept of fraction tree is introduced. This has profound consequences both in mathematical sub-disciplines and other branches science technology. we shall witness connection with that Fibonacci sequence relate terms continued fractions.

Protein-fragment seqlets typically feature about 10 amino acid residue positions that are fixed to within conservative substitutions but usually separated by a number of prescribed gaps with arbitrary residue content. By quantifying a general amino acid residue sequence in terms of the associated codon number sequence, we have found a precise modular Fibonacci sequence in a continuous gap-free ...