Problem p03766 - Generation 1

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), representing some quantity or size.

### 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! Below is a test case generator for the given problem statement.

### Test Case Generator:

```python
import random

def tcgen():
    # Generate a random value for n within the given constraints
    n = random.randint(1, 10**6)
    
    return n

# Example usage:
# Generate a test case
n = tcgen()
print(n)
```

### Explanation:

This test case generator creates a random test case for the provided problem statement:

1. **n**: Randomly generated integer between 1 and 10^6.

You can run this generator multiple times to create different test cases for the problem.

### Example Output from the Generator:

```plaintext
834057
```

This output corresponds to a single randomly generated test case with \( n \) as 834057. You can use this in your program to test the correctness and performance of your solution.

Extract Arguments

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