def squares(n):
    """Compute the squares of numbers from 1 to n, such that the 
    ith element of the returned list equals i^2.
    
    """
    ### BEGIN SOLUTION
    if n < 1:
        raise ValueError("n must be greater than or equal to 1")
    return [i ** 2 for i in range(1, n + 1)]
    ### END SOLUTION

"""
Explanation: For this problem set, we'll be using the Jupyter notebook:


Part A (2 points)
Write a function that returns a list of numbers, such that $x_i=i^2$, for $1\leq i \leq n$. Make sure it handles the case where $n<1$ by raising a ValueError.
End of explanation
"""


squares(10)

"""Check that squares returns the correct output for several inputs"""
assert squares(1) == [1]
assert squares(2) == [1, 4]
assert squares(10) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
assert squares(11) == [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121]

"""Check that squares raises an error for invalid inputs"""
try:
    squares(0)
except ValueError:
    pass
else:
    raise AssertionError("did not raise")

try:
    squares(-4)
except ValueError:
    pass
else:
    raise AssertionError("did not raise")

"""
Explanation: Your function should print [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] for $n=10$. Check that it does:
End of explanation
"""


def sum_of_squares(n):
    """Compute the sum of the squares of numbers from 1 to n."""
    ### BEGIN SOLUTION
    return sum(squares(n))
    ### END SOLUTION

"""
Explanation: Part B (1 point)
Using your squares function, write a function that computes the sum of the squares of the numbers from 1 to $n$. Your function should call the squares function -- it should NOT reimplement its functionality.
End of explanation
"""


sum_of_squares(10)

"""Check that sum_of_squares returns the correct answer for various inputs."""
assert sum_of_squares(1) == 1
assert sum_of_squares(2) == 5
assert sum_of_squares(10) == 385
assert sum_of_squares(11) == 506

"""Check that sum_of_squares relies on squares."""
orig_squares = squares
del squares
try:
    sum_of_squares(1)
except NameError:
    pass
else:
    raise AssertionError("sum_of_squares does not use squares")
finally:
    squares = orig_squares

"""
Explanation: The sum of squares from 1 to 10 should be 385. Verify that this is the answer you get:
End of explanation
"""


def pyramidal_number(n):
    """Returns the n^th pyramidal number"""
    return sum_of_squares(n)

"""
Explanation: Part C (1 point)
Using LaTeX math notation, write out the equation that is implemented by your sum_of_squares function.
$\sum_{i=1}^n i^2$

Part D (2 points)
Find a usecase for your sum_of_squares function and implement that usecase in the cell below.
End of explanation
"""


