



"""simple docstring"""




import numpy as np



def 	UpperCAmelCase__   (				A__   )      ->      np.array:




   """simple docstring"""
   return 1 / (1 + np.exp(-vector   ))


if __name__ == "__main__":
     import doctest

     doctest.testmod()







"""simple docstring"""




def 	UpperCAmelCase__   (				A__   )      ->      list[int]:




   """simple docstring"""

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


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




