Related papers: Record values in appending and prepending bitstrin…
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a…
We give three different computations of the total number of runs of length $i$ in binary $n$-strings, and we discuss the connection of this problem with the compositions of $n$.
We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same…
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the…
We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this…
When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to…
We design, implement and test a simple algorithm which computes the approximate entropy of a finite binary string of arbitrary length. The algorithm uses a weighted average of the Shannon Entropies of the string and all but the last binary…
The binary string matching problem consists in finding all the occurrences of a pattern in a text where both strings are built on a binary alphabet. This is an interesting problem in computer science, since binary data are omnipresent in…
Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers,…
An algorithm counting the number of ones in a binary word is presented running in time $O(\log\log b)$ where $b$ is the number of ones. The operations available include bit-wise logical operations and multiplication.
Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results…
The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a…
We determine the average number of distinct subsequences in a random binary string, and derive an estimate for the average number of distinct subsequences of a particular length.
We show that the number of short binary signed-digit representations of an integer $n$ is equal to the $n$-th term in the Stern sequence. Various proofs are provided, including direct, bijective, and generating function proofs. We also show…
If the list of binary numbers is read by upward-sloping diagonals, the resulting ``sloping binary numbers'' 0, 11, 110, 101, 100, 1111, 1010, ... (or 0, 3, 6, 5, 4, 15, 10, ...) have some surprising properties. We give formulae for the n-th…
The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded…
We define the ``shift-match number'' for a binary string and we compute the probability of occurrence of a given string as a subsequence in longer strings in terms of its shift-match number. We thus prove that the string matching…
This paper introduces a binary encoding that supports arbitrarily large, small and precise decimals. It completely preserves information and order. It does not rely on any arbitrary use-case-based choice of calibration and is readily…