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Calculate number of 1-runs in a binary string expression

Source: Bangladesh Mathematical Olympiad 2021 Problem 7

February 26, 2022
combinatoricsnumber theory

Problem Statement

A binary string is a word containing only 00s and 11s. In a binary string, a 1−1-run is a non extendable substring containing only 11s. Given a positive integer nn, let B(n)B(n) be the number of 1−1-runs in the binary representation of nn. For example, B(107)=3B(107)=3 since 107107 in binary is 11010111101011 which has exactly three 1−1-runs. What is the following expression equal to? B(1)+B(2)+B(3)+⋯+B(255)B(1)+B(2)+B(3)+ \dots + B(255)
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