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
To generate test cases for the given problem, we need to consider the constraints and ensure that the generated test cases cover the edge cases, typical cases, and large input cases effectively. Here's how you can write a test case generator for the problem statement:
```python
import random
def tcgen():
# Generate a random integer n within the given constraints
n = random.randint(1, 10**6)
return n
# Example usage:
if __name__ == "__main__":
for _ in range(10): # Generate 10 test cases
print(tcgen())
```
This function generates a random integer \( n \) between 1 and 1,000,000, which is the constraint given in the problem statement. You can generate multiple test cases by invoking this function multiple times.
Here are some points to ensure while generating test cases:
1. Ensure that your test cases include both small values (e.g., 1, 2, etc.) and large values (close to 1,000,000).
2. Ensure some edge cases like the minimum (1) and the maximum (1,000,000) are included.
Given the nature of the problem where the output is based on the sequences that follow specific rules, the generated test cases are simply different values of \( n \) within the constraints.
You might also want to include some specific test cases manually, if needed, for thorough testing. For instance:
- Minimum edge case: `1`
- Small value: `2`
- Large value: `999999`
- Maximum edge case: `1000000`
This will help ensure that you are covering a wide range of possible inputs when testing your solution.
Extract Arguments
def extract_arguments(fh):
n = int(fh.readline().strip())
return n