





def      lowerCAmelCase_ (       __a = 1000		)     ->		int:







  """simple docstring"""





  return sum(2 * a * ((a - 1) // 2) for a in range(3	,    n + 1		)		)


if __name__ == "__main__":
     print(solution())








from typing import Any





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		list:







  """simple docstring"""





  _validation(
      __a	,    __a	,    __a	,    __a	,    __a	,    )
  # Creates data structures and fill initial step
  lowerCamelCase__:      dict			  ={}
  lowerCamelCase__:      dict			  ={}
  for state in states_space:
    lowerCamelCase__:      Optional[Any]			  =observations_space[0]
    lowerCamelCase__:      List[Any]			  =(
        initial_probabilities[state] * emission_probabilities[state][observation]
    )
    lowerCamelCase__:      int			  =None

  # Fills the data structure with the probabilities of
  # different transitions and pointers to previous states
  for o in range(1	,    len(__a		)		):
    lowerCamelCase__:      Tuple			  =observations_space[o]
    lowerCamelCase__:      Optional[Any]			  =observations_space[o - 1]
    for state in states_space:
      # Calculates the argmax for probability function
      lowerCamelCase__:      Tuple			  =""
      lowerCamelCase__:      Optional[Any]			  =-1
      for k_state in states_space:
        lowerCamelCase__:      int			  =(
            probabilities[(k_state, prior_observation)]
            * transition_probabilities[k_state][state]
            * emission_probabilities[state][observation]
        )
        if probability > max_probability:
          lowerCamelCase__:      List[str]			  =probability
          lowerCamelCase__:      int			  =k_state

            # Update probabilities and pointers dicts
      lowerCamelCase__:      Any			  =(
          probabilities[(arg_max, prior_observation)]
          * transition_probabilities[arg_max][state]
          * emission_probabilities[state][observation]
      )

      lowerCamelCase__:      int			  =arg_max

    # The final observation
  lowerCamelCase__:      Any			  =observations_space[len(__a		) - 1]

  # argmax for given final observation
  lowerCamelCase__:      Optional[Any]			  =""
  lowerCamelCase__:      int			  =-1
  for k_state in states_space:
    lowerCamelCase__:      Tuple			  =probabilities[(k_state, final_observation)]
    if probability > max_probability:
      lowerCamelCase__:      List[Any]			  =probability
      lowerCamelCase__:      Dict			  =k_state
  lowerCamelCase__:      str			  =arg_max

  # Process pointers backwards
  lowerCamelCase__:      Union[str, Any]			  =last_state
  lowerCamelCase__:      List[str]			  =[]
  for o in range(len(__a		) - 1	,    -1	,    -1		):
    result.append(__a		)
    lowerCamelCase__:      Union[str, Any]			  =pointers[previous, observations_space[o]]
  result.reverse()

  return result





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





  _validate_not_empty(
      __a	,    __a	,    __a	,    __a	,    __a	,    )
  _validate_lists(__a	,    __a		)
  _validate_dicts(
      __a	,    __a	,    __a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





  if not all(
      [
          observations_space,
          states_space,
          initial_probabilities,
          transition_probabilities,
          emission_probabilities,
      ]		):
    raise ValueError("There's an empty parameter"		)





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  _validate_list(__a	,    "observations_space"		)
  _validate_list(__a	,    "states_space"		)





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  if not isinstance(_object	,    __a		):
    lowerCamelCase__:      Tuple			  =F"""{var_name} must be a list"""
    raise ValueError(__a		)
  else:
    for x in _object:
      if not isinstance(__a	,    __a		):
        lowerCamelCase__:      str			  =F"""{var_name} must be a list of strings"""
        raise ValueError(__a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    )     ->		None:







  """simple docstring"""





  _validate_dict(__a	,    "initial_probabilities"	,    __a		)
  _validate_nested_dict(__a	,    "transition_probabilities"		)
  _validate_nested_dict(__a	,    "emission_probabilities"		)





def      lowerCAmelCase_ (       __a	,    __a		)     ->		None:







  """simple docstring"""





  _validate_dict(_object	,    __a	,    __a		)
  for x in _object.values():
    _validate_dict(__a	,    __a	,    __a	,    __a		)





def      lowerCAmelCase_ (       __a	,    __a	,    __a	,    __a = False		)     ->		None:







  """simple docstring"""





  if not isinstance(_object	,    __a		):
    lowerCamelCase__:      Optional[int]			  =F"""{var_name} must be a dict"""
    raise ValueError(__a		)
  if not all(isinstance(__a	,    __a		) for x in _object		):
    lowerCamelCase__:      Tuple			  =F"""{var_name} all keys must be strings"""
    raise ValueError(__a		)
  if not all(isinstance(__a	,    __a		) for x in _object.values()		):
    lowerCamelCase__:      Dict			  ="nested dictionary " if nested else ""
    lowerCamelCase__:      List[str]			  =F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
    raise ValueError(__a		)


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

     testmod()

