Formatted question description: https://leetcode.ca/all/1680.html

# 1680. Concatenation of Consecutive Binary Numbers (Medium)

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

Related Topics:
Math

## Solution 1.

Let f(n) be the answer. sum_len(a, b) = sum( len(i) | a <= i <= b) and len(i) is the number of bits in i.

For example: len(1) = 1, len(3) = 2, len(6) = 3. sum_len(1,4) = len(1) + len(2) + len(3) + len(4) = 1 + 2 + 2 + 3 = 8.

So we have

f(n)   = (1 << sum_len(2, n))   + (2 << sum_len(3, n))   + ... + ((n - 1) << sum_len(n, n)) + (n << 0)

// Example: f(4) = 11011100 = (1 << (2+2+3)) + (2 << (2+3)) + (3 << 3) +(4 << 0)

f(n-1) = (1 << sum_len(2, n-1)) + (2 << sum_len(3, n-1)) + ... + ((n - 1) << 0)

f(n) = (f(n-1) << len(n)) + n


Since f(0) = 0, we can iteratively build f(n).

f(0) = 0
f(1) = 1     = (f(0) << 1) + 1  // len(1) = 1
f(2) = 110   = (f(1) << 2) + 2  // len(2) = 2
f(3) = 11011 = (f(2) << 2) + 3  // len(3) = 2
...

// OJ: https://leetcode.com/problems/concatenation-of-consecutive-binary-numbers/

// Time: O(NlogN)
// Space: O(1)
class Solution {
public:
int concatenatedBinary(int n) {
long ans = 0, mod = 1e9+7;
for (int i = 1; i <= n; ++i) {
int len = 0;
for (int j = i; j; j >>= 1, ++len);
ans = ((ans << len) % mod + i) % mod;
}
return ans;
}
};


## Solution 2.

We spent O(logN) time for calculating the len. We can reduce it to O(1) with the help of __builtin_clz which returns the number of leading zeros for a number, so len = 32 - __builtin_clz(i).

// OJ: https://leetcode.com/problems/concatenation-of-consecutive-binary-numbers/

// Time: O(N)
// Space: O(1)
class Solution {
public:
int concatenatedBinary(int n) {
long ans = 0, mod = 1e9+7;
for (int i = 1; i <= n; ++i) ans = ((ans << (32 - __builtin_clz(i))) % mod + i) % mod;
return ans;
}
};


Or, with the observation that the len only increment when the i is a power of 2, we can increment len only when i has a single bit 1. We can check this via (i & (i - 1)) == 0 or __builtin_popcount(i) == 1.

// OJ: https://leetcode.com/problems/concatenation-of-consecutive-binary-numbers/

// Time: O(N)
// Space: O(1)
class Solution {
public:
int concatenatedBinary(int n) {
long ans = 0, mod = 1e9+7, len = 0;
for (int i = 1; i <= n; ++i) {
if ((i & (i - 1)) == 0) ++len;
ans = ((ans << len) % mod + i) % mod;
}
return ans;
}
};


Java

class Solution {
public int concatenatedBinary(int n) {
final int MODULO = 1000000007;
long num = 0;
int bits = 0;
for (int i = 1; i <= n; i++) {
if ((i & (i - 1)) == 0)
bits++;
num = ((num << bits) + i) % MODULO;
}
return (int) num;
}
}