



"""simple docstring"""







from __future__ import annotations

from typing import Any
class        __A       :







		'''simple docstring'''





		def __init__(	self						:  Optional[Any]			,_snake_case						:  int			,_snake_case						:  int			,_snake_case						:  float = 0						)					->		None:






				"""simple docstring"""



				lowercase__		,			lowercase__   :    int				=			row, column
				lowercase__   :    Any				=			[[default_value for c in range(_snake_case						)] for r in range(_snake_case						)]







		def __str__(	self						:  Dict						)					->		str:






				"""simple docstring"""



				lowercase__   :    List[str]				=			f"""Matrix consist of {self.row} rows and {self.column} columns\n"""

				# Make string identifier
				lowercase__   :    Optional[int]				=			0
				for row_vector in self.array:
						for obj in row_vector:
								lowercase__   :    str				=			max(_snake_case			,len(str(_snake_case						)						)						)
				lowercase__   :    Tuple				=			f"""%{max_element_length}s"""

				# Make string and return
				def single_line(_snake_case						:  list[float]						) -> str:
						nonlocal string_format_identifier
						lowercase__   :    Dict				=			'''['''
						line += ", ".join(string_format_identifier % (obj,) for obj in row_vector						)
						line += "]"
						return line

				s += "\n".join(single_line(_snake_case						) for row_vector in self.array						)
				return s







		def __repr__(	self						:  Optional[Any]						)					->		str:






				"""simple docstring"""



				return str(self						)







		def 					UpperCAmelCase    (	self						:  Tuple			,_snake_case						:  tuple[int, int]						)					->		bool:






				"""simple docstring"""



				if not (isinstance(_snake_case			,(list, tuple)						) and len(_snake_case						) == 2):
						return False
				elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
						return False
				else:
						return True







		def __getitem__(	self						:  Optional[int]			,_snake_case						:  tuple[int, int]						)					->		Any:






				"""simple docstring"""



				assert self.validate_indicies(_snake_case						)
				return self.array[loc[0]][loc[1]]







		def __setitem__(	self						:  Union[str, Any]			,_snake_case						:  tuple[int, int]			,_snake_case						:  float						)					->		None:






				"""simple docstring"""



				assert self.validate_indicies(_snake_case						)
				lowercase__   :    Any				=			value







		def __add__(	self						:  int			,_snake_case						:  Matrix						)					->		Matrix:






				"""simple docstring"""



				assert isinstance(_snake_case			,_snake_case						)
				assert self.row == another.row and self.column == another.column

				# Add
				lowercase__   :    List[Any]				=			Matrix(self.row			,self.column						)
				for r in range(self.row						):
						for c in range(self.column						):
								lowercase__   :    Any				=			self[r, c] + another[r, c]
				return result







		def __neg__(	self						:  Dict						)					->		Matrix:






				"""simple docstring"""



				lowercase__   :    Dict				=			Matrix(self.row			,self.column						)
				for r in range(self.row						):
						for c in range(self.column						):
								lowercase__   :    List[str]				=			-self[r, c]
				return result







		def __sub__(	self						:  int			,_snake_case						:  Matrix						)					->		Matrix:






				"""simple docstring"""



				return self + (-another)







		def __mul__(	self						:  int			,_snake_case						:  int | float | Matrix						)					->		Matrix:






				"""simple docstring"""



				if isinstance(_snake_case			,(int, float)						):  # Scalar multiplication
						lowercase__   :    str				=			Matrix(self.row			,self.column						)
						for r in range(self.row						):
								for c in range(self.column						):
										lowercase__   :    Tuple				=			self[r, c] * another
						return result
				elif isinstance(_snake_case			,_snake_case						):  # Matrix multiplication
						assert self.column == another.row
						lowercase__   :    Any				=			Matrix(self.row			,another.column						)
						for r in range(self.row						):
								for c in range(another.column						):
										for i in range(self.column						):
												result[r, c] += self[r, i] * another[i, c]
						return result
				else:
						lowercase__   :    Union[str, Any]				=			f"""Unsupported type given for another ({type(_snake_case						)})"""
						raise TypeError(_snake_case						)







		def 					UpperCAmelCase    (	self						:  Dict						)					->		Matrix:






				"""simple docstring"""



				lowercase__   :    int				=			Matrix(self.column			,self.row						)
				for r in range(self.row						):
						for c in range(self.column						):
								lowercase__   :    Dict				=			self[r, c]
				return result







		def 					UpperCAmelCase    (	self						:  int			,_snake_case						:  Matrix			,_snake_case						:  Matrix						)					->		Any:






				"""simple docstring"""



				assert isinstance(_snake_case			,_snake_case						) and isinstance(_snake_case			,_snake_case						)
				assert self.row == self.column == u.row == v.row  # u, v should be column vector
				assert u.column == v.column == 1  # u, v should be column vector

				# Calculate
				lowercase__   :    Dict				=			v.transpose()
				lowercase__   :    Union[str, Any]				=			(v_t * self * u)[0, 0] + 1
				if numerator_factor == 0:
						return None  # It's not invertable
				return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":



							def      __UpperCAmelCase      (    )							->      None:
									# a^(-1)
									lowercase__   :    Tuple				=			Matrix(3						,     3						,     0	)
									for i in range(3	):
											lowercase__   :    List[Any]				=			1
									print(f"""a^(-1) is {ainv}"""	)
									# u, v
									lowercase__   :    Tuple				=			Matrix(3						,     1						,     0	)
									lowercase__		,			lowercase__		,			lowercase__   :    Dict				=			1, 2, -3
									lowercase__   :    List[Any]				=			Matrix(3						,     1						,     0	)
									lowercase__		,			lowercase__		,			lowercase__   :    str				=			4, -2, 5
									print(f"""u is {u}"""	)
									print(f"""v is {v}"""	)
									print(f"""uv^T is {u * v.transpose()}"""	)
									# Sherman Morrison
									print(f"""(a + uv^T)^(-1) is {ainv.sherman_morrison(__lowerCamelCase						,     __lowerCamelCase	)}"""	)



							def      __UpperCAmelCase      (    )							->      None:
									import doctest

									doctest.testmod()





							testa()





"""simple docstring"""







def      __UpperCAmelCase      (    __lowerCamelCase = 50	)							->      int:
		lowercase__   :    int				=			[[0] * 3 for _ in range(length + 1	)]

		for row_length in range(length + 1	):
				for tile_length in range(2						,     5	):
						for tile_start in range(row_length - tile_length + 1	):
								different_colour_ways_number[row_length][tile_length - 2] += (
								    different_colour_ways_number[row_length - tile_start - tile_length][
								        tile_length - 2
								    ]
								    + 1
								)

		return sum(different_colour_ways_number[length]	)


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