Description

Given an integer n, return the decimal value of the binary string formed by concatenating the binary representations of 1 to n in order, modulo 109 + 7.

Example 1:

Input: n = 1 Output: 1 Explanation: “1” in binary corresponds to the decimal value 1. Example 2:

Input: n = 3 Output: 27 Explanation: In binary, 1, 2, and 3 corresponds to “1”, “10”, and “11”. After concatenating them, we have “11011”, which corresponds to the decimal value 27. Example 3:

Input: n = 12 Output: 505379714 Explanation: The concatenation results in “1101110010111011110001001101010111100”. The decimal value of that is 118505380540. After modulo 109 + 7, the result is 505379714.

Constraints:

1 <= n <= 105

Solution

The idea is if we know the answer for n -1, we can construct the answer for n. Let the answer for n - 1 be f(n-1). Then f(n) = f(n-1)*2^d+n where d is number of digit in binary representation of n.

We can write this recursively. Since it is tail recursion, we can rewrite it iteratively. Below is the implementation

class Solution:
    def concatenatedBinary(self, n: int) -> int:
        mod = 1000000000 + 7
        ret = 0
        for i in range(1, n + 1):
            t = i
            count = 0
            while t > 0:
                t >>= 1
                count += 1
            ret = ((1 << count) * ret + i) % mod
        return ret