recurrent sequence

Methods of Correcting Errors in Messages Encoded by Fibonacci Matrices

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.

FROM NEWTON'S BINOMIAL AND PASCAL’S TRIANGLE TO СOLLATZ'S PROBLEM

It is shown that: 1. The sequence {20,21, 22, 23, 24, 25, 26, 27,28,...} that forms the main graph m=1 of Collatz is related to the power transformation of Newton's binomial (1+1)ξ, ξ=0, 1, 2, 3,... 2. The main Kmain and side m >1 graphs and their corresponding sequences {Kmain } and {Km } are related by the relation {Km }=m⋅{Kmain }. 3.

RECURRENCE AND STRUCTURING OF SEQUENCES OF TRANSFORMATIONS 3N + 1 AS ARGUMENTS FOR CONFIRMATION OF THE СOLLATZ HYPOTHESIS

It is shown that infinites of the subsequence of odd numbers is not a counterargument of the violation of the Collatz hypothesis, but a universal characteristic of transformations of natural numbers by the 3n + 1 algorithm. A recurrent relationship is established between the parameters of the sequence of Collatz transformations of an arbitrary pair of natural numbers n and 2n.