


"""simple docstring"""





from __future__ import annotations





def     lowercase__	(  lowerCamelCase     :  list				,	lowerCamelCase     :  list				)					->     list:
  lowerCAmelCase__    :    Dict								=					[None] * (len(lowerCamelCase				) + len(lowerCamelCase				))

  lowerCAmelCase__    :    Optional[Any]								=					0  # pointer to current index for left Half
  lowerCAmelCase__    :    Any								=					0  # pointer to current index for the right Half
  lowerCAmelCase__    :    Dict								=					0  # pointer to current index for the sorted array Half

  while pointera < len(lowerCamelCase				) and pointera < len(lowerCamelCase				):
    if left_half[pointera] < right_half[pointera]:
      lowerCAmelCase__    :    Tuple								=					left_half[pointera]
      pointera += 1
      index += 1
    else:
      lowerCAmelCase__    :    int								=					right_half[pointera]
      pointera += 1
      index += 1
  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Tuple								=					left_half[pointera]
    pointera += 1
    index += 1

  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Optional[int]								=					right_half[pointera]
    pointera += 1
    index += 1

  return sorted_array





def     lowercase__	(  lowerCamelCase     :  list				)					->     list:
  if len(lowerCamelCase				) <= 1:
    return array
  # the actual formula to calculate the middle element = left + (right - left) // 2
  # this avoids integer overflow in case of large N
  lowerCAmelCase__    :    Optional[int]								=					0 + (len(lowerCamelCase				) - 0) // 2

  # Split the array into halves till the array length becomes equal to One
  # merge the arrays of single length returned by mergeSort function and
  # pass them into the merge arrays function which merges the array
  lowerCAmelCase__    :    Union[str, Any]								=					array[:middle]
  lowerCAmelCase__    :    List[str]								=					array[middle:]

  return merge(merge_sort(lowerCamelCase				)				,	merge_sort(lowerCamelCase				)				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()




"""simple docstring"""





def     lowercase__	(  lowerCamelCase     :  int				)					->     int:
  if not isinstance(lowerCamelCase				,	lowerCamelCase				):
    raise TypeError("recursive_lucas_number accepts only integer arguments."				)
  if n_th_number == 0:
    return 2
  if n_th_number == 1:
    return 1

  return recursive_lucas_number(n_th_number - 1				) + recursive_lucas_number(
      n_th_number - 2				)





def     lowercase__	(  lowerCamelCase     :  int				)					->     int:
  if not isinstance(lowerCamelCase				,	lowerCamelCase				):
    raise TypeError("dynamic_lucas_number accepts only integer arguments."				)
  lowerCAmelCase__				,		lowerCAmelCase__    :    str								=					2, 1
  for _ in range(lowerCamelCase				):
    lowerCAmelCase__				,		lowerCAmelCase__    :    str								=					b, a + b
  return a


if __name__ == "__main__":
   from doctest import testmod

   testmod()
   __UpperCAmelCase    				=      int(input("Enter the number of terms in lucas series:\n").strip())
   print("Using recursive function to calculate lucas series:")
   print(" ".join(str(recursive_lucas_number(i)) for i in range(n)))
   print("\nUsing dynamic function to calculate lucas series:")
   print(" ".join(str(dynamic_lucas_number(i)) for i in range(n)))
