
'''simple docstring'''



def 	_A						(							lowercase__		):
						return n * (2 * n - 1)





def 	_A						(							lowercase__		):
						lowercase__								=    (1 + 24 * n) ** 0.5
						return ((1 + root) / 6) % 1 == 0





def 	_A						(							lowercase__ = 144		):
						lowercase__								=    start
						lowercase__								=    hexagonal_num(_lowerCAmelCase		)
						while not is_pentagonal(_lowerCAmelCase		):
												n += 1
												lowercase__								=    hexagonal_num(_lowerCAmelCase		)
						return num


if __name__ == "__main__":
	print(F'''{solution()} = ''')








'''simple docstring'''



def 	_A						(							lowercase__		):
						if not isinstance(lowercase__						,						lowercase__		):
												raise ValueError("""Input series is not valid, valid series - [1, 2/3, 2]"""		)
						if len(lowercase__		) == 0:
												raise ValueError("""Input list must be a non empty list"""		)
						if len(lowercase__		) == 1 and series[0] != 0:
												return True
						lowercase__								=    []
						lowercase__								=    len(lowercase__		)
						for i in range(0						,						lowercase__		):
												if series[i] == 0:
																		raise ValueError("""Input series cannot have 0 as an element"""		)
												rec_series.append(1 / series[i]		)
						lowercase__								=    rec_series[1] - rec_series[0]
						for index in range(2						,						lowercase__		):
												if rec_series[index] - rec_series[index - 1] != common_diff:
																		return False
						return True





def 	_A						(							lowercase__		):
						if not isinstance(lowercase__						,						lowercase__		):
												raise ValueError("""Input series is not valid, valid series - [2, 4, 6]"""		)
						if len(lowercase__		) == 0:
												raise ValueError("""Input list must be a non empty list"""		)
						lowercase__								=    0
						for val in series:
												answer += 1 / val
						return len(lowercase__		) / answer


if __name__ == "__main__":
				import doctest

				doctest.testmod()






