Observe that the binary string is formed by concatenating the binary representations of numbers from 1 to n. How can you compute the decimal value of this binary string efficiently without explicitly constructing the entire string?

Consider how the decimal value changes when you append a new binary number to the existing string. Can you derive a mathematical relationship between the current decimal value and the next binary number to be appended?

The key insight is to recognize that appending a binary number `k` to a binary string `s` is equivalent to `(s * 2^len(k)) + k`. Use this to update the decimal value iteratively for each number from 1 to n, while keeping track of the length of the binary string so far.