Zeeshan, a budding programmer with a penchant for puzzles, found himself immersed in a curious challenge. The task was to determine the sum of consecutive natural numbers, starting from 1, up to a specific count. This count was peculiar, being equivalent to the total number of distinct binary strings of length 'n'. For instance, if 'n' were 3, the possible bit strings (000, 001, ..., 111), and the desired sum would encompass numbers from 1 to the number of distinct Binary String length.

An Integer representing the Sum of numbers of the distinct binary string length