从一个元素开始，将函数f重复应用于每个新元素，同时将它们添加到向量中

def f(x):
    # 假设 f 是一个简单的函数，返回 x 的平方
    return x * x

def apply_f_to_vector(initial_element, num_iterations):
    vector = [initial_element]
    for _ in range(num_iterations):
        new_element = f(vector[-1])
        vector.append(new_element)
    return vector

# 示例使用
initial_element = 2
num_iterations = 5
result = apply_f_to_vector(initial_element, num_iterations)
print(result)  # 输出: [2, 4, 16, 256, 65536, 4294967296]

def apply_f_to_vector_iterative(initial_element, num_iterations):
    vector = [initial_element]
    current_element = initial_element
    for _ in range(num_iterations):
        current_element = f(current_element)
        vector.append(current_element)
    return vector

def f(x, memo={}):
    if x not in memo:
        memo[x] = x * x
    return memo[x]