Two problems about calculating the number of palindromes with different numbers of digits

The aim of this work is to obtain two types of regularities for calculating the number of palindromes. The main results of the work are presented by solving two problems. In the first problem describes how to find the total number of different locations of palindromes in binary digits that do not exceed the given numbers of digits. The second problem is devoted to determining the number of palindromes of an arbitrary number system that have the same numbers of digits. The theoretical results of the work are accompanied by examples of the application of the described regularities to obtain the total number of locations of the binary palindromes with the number of digits that do not exceed the specified, as well as to calculate the number of palindromes of different number systems with predetermined numbers of digits. The considered regularities are supposed to be applied in the analysis of numerical and computer data.