Fibonacci numbers

The main problems of detection and available methods of correcting errors in encoded messages with Fibonacci matrices, which make it possible to find and correct one, two and three errors in the same or different lines of the code word, are analyzed. It has been found that even in the last decade, many scientists have published a significant number of various publications, each of which to one degree or another substantiates the expediency of using Fibonacci matrices for (de)coding data.

In this paper, it is shown that the Fibonacci triangle is formed from the elements of power transformations of a quadratic trinomial. It is binary structured by domains of rows of equal lengths, in which the sum of numbers forms a sequence of certain numbers. This sequence coincides with the transformed bisection of the classical sequence of Fibonacci numbers. The paper substantiates Pascal's rule for calculating elements in the lines of a Fibonacci triangle. The general relations of two forgings of numbers in lines of a triangle of Fibonacci for arbitrary values are received

This paper studies regularities of proportional division, on the basis of which we show the possibility of effective application of the golden section method to modeling regularities of atomic systems and positioning of elements of noble gases of the periodic table. It is illustrated that by partial reconstruction of the Mendeleev tables, the elements of noble gases can be arranged along lines whose slope tangents in the coordinate system “the atomic number – the relative atomic mass” are in close agreement with the sequence of inverse Fibonacci numbers.