> For the complete documentation index, see [llms.txt](https://gl01.gitbook.io/gfg-editorials/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://gl01.gitbook.io/gfg-editorials/2024/06-2024-june-2/03-trail-of-ones.md).

# 03. Trail of ones

The problem can be found at the following link: [Question Link](https://www.geeksforgeeks.org/problems/trail-of-ones3242/1)

## My Approach

* Initialize `mod` to (10^9 + 7) to handle large numbers and prevent overflow.
* Initialize `out` to 1, which will store the number of valid sequences of ones.
* Initialize `x` to 0 and `y` to 1, representing the number of valid sequences ending in a single one or two consecutive ones respectively.
* Iterate from 3 to `n`:
  * Update `out` to `(out * 2 + x + y) % mod` to include new sequences formed by appending "0" or "1" to previous valid sequences.
  * Temporarily store the current value of `x` in `z`.
  * Update `x` to the current value of `y`.
  * Update `y` to `(x + z) % mod`, maintaining the count of sequences ending in two consecutive ones.
* Return the final value of `out`.

## Time and Auxiliary Space Complexity

* **Time Complexity**: `(O(n))`, since we iterate from 3 to `n`.
* **Auxiliary Space Complexity**: `(O(1))`, as we use a constant amount of extra space regardless of the input size.

## Code (C++)

```cpp
class Solution {
public:
    int numberOfConsecutiveOnes(int n) {
        int mod = 1e9 + 7;
        long out = 1;
        int x = 0, y = 1;
        for (int i = 3; i <= n; ++i) {
            out = (out * 2 + x + y) % mod;
            int z = x;
            x = y;
            y = (x + z) % mod;
        }
        return out;
    }
};
```

## Contribution and Support

For discussions, questions, or doubts related to this solution, please visit our [discussion section](https://github.com/getlost01/gfg-potd/discussions). We welcome your input and aim to foster a collaborative learning environment.

If you find this solution helpful, consider supporting us by giving a `⭐ star` to the [getlost01/gfg-potd](https://github.com/getlost01/gfg-potd) repository.


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