





def  SCREAMING_SNAKE_CASE				(      snake_case							)		->    list[int]:

		if length <= 0 or not isinstance(snake_case						,   snake_case							):
				raise ValueError('Length must be a positive integer.'							)
		return [n * (2 * n - 1) for n in range(snake_case							)]


if __name__ == "__main__":
		print(hexagonal_numbers(length=5))
		print(hexagonal_numbers(length=10))











def  SCREAMING_SNAKE_CASE				(      snake_case						,   snake_case						,   snake_case							)		->    int:

		def count_of_possible_combinations(snake_case							) -> int:
				if target < 0:
						return 0
				if target == 0:
						return 1
				return sum(count_of_possible_combinations(target - item							) for item in array							)

		return count_of_possible_combinations(snake_case							)



def  SCREAMING_SNAKE_CASE				(      snake_case						,   snake_case						,   snake_case							)		->    int:

		def count_of_possible_combinations_with_dp_array(
		    snake_case						,   snake_case							) -> int:
				if target < 0:
						return 0
				if target == 0:
						return 1
				if dp_array[target] != -1:
						return dp_array[target]
				__lowercase											=     sum(
				    count_of_possible_combinations_with_dp_array(target - item						,   snake_case							)
				    for item in array							)
				__lowercase											=     answer
				return answer

		__lowercase											=     [-1] * (target + 1)
		return count_of_possible_combinations_with_dp_array(snake_case						,   snake_case							)



def  SCREAMING_SNAKE_CASE				(      snake_case						,   snake_case						,   snake_case							)		->    int:
		__lowercase											=     [0] * (target + 1)
		__lowercase											=     1

		for i in range(1						,   target + 1							):
				for j in range(snake_case							):
						if i - array[j] >= 0:
								dp_array[i] += dp_array[i - array[j]]

		return dp_array[target]


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		SCREAMING_SNAKE_CASE_   :      Tuple			=   3
		SCREAMING_SNAKE_CASE_   :      Dict			=   5
		SCREAMING_SNAKE_CASE_   :      Any			=   [1, 2, 5]
		print(combination_sum_iv(n, array, target))




