Problem p03766 - Generation 2

Orig Description

Score : 1000 points
Problem StatementHow many infinite sequences a_1, a_2, ... consisting of {{1, ... ,n}} satisfy the following conditions?
The n-th and subsequent elements are all equal. That is, if n \leq i,j, a_i = a_j.
For every integer i, the a_i elements immediately following the i-th element are all equal. That is, if i < j < k\leq i+a_i, a_j = a_k.
Find the count modulo 10^9+7.
Constraints
1 \leq n \leq 10^6
InputInput is given from Standard Input in the following format:
n
OutputPrint how many sequences satisfy the conditions, modulo 10^9+7.
Sample Input 12
Sample Output 14
The four sequences that satisfy the conditions are:
1, 1, 1, ...
1, 2, 2, ...
2, 1, 1, ...
2, 2, 2, ...
Sample Input 2654321
Sample Output 2968545283

Extracted Specification

An integer n (1 ≤ n ≤ 10^6).

### Example Input:

```
12
```

### Function Signature:
Write a function f(n) that takes in the input. 
def f(n: int):
    ''' 
    n: an integer 
    '''

Test Case Generator

Certainly! Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Randomly generate N within the constraints 1 ≤ n ≤ 10^6
    N = random.randint(1, 10**6)
    
    return N

# Example of using the test case generator
if __name__ == "__main__":
    # Generate a test case
    N = tcgen()
    
    # Print the test case in the required format
    print(N)
```

This generator will produce a single integer \( N \) in the range from 1 to \( 10^6 \), which is the input according to the problem statement. You can run this generator to produce random test cases for your competitive programming problem. 

For instance, running the generator might yield an output like:
```
857321
```

This output represents a test case where \( n = 857321 \).

Extract Arguments

def extract_arguments(fh):
    n = int(fh.readline().strip())
    return (n,)