




import numpy as np
from numpy import floataa
from numpy.typing import NDArray


def   A		(__A  :       List[Any]  ,							__A  :       int			)  -> Optional[int]:







     """simple docstring"""


     UpperCAmelCase_		=				np.shape(a_			)

     UpperCAmelCase_		=				np.zeros((rows, 1)  ,							dtype=a_			)
     for row in reversed(range(a_			)			):
          UpperCAmelCase_		=				0
          for col in range(row + 1  ,							a_			):
               total += coefficients[row, col] * x[col]

          UpperCAmelCase_		=				(vector[row] - total) / coefficients[row, row]

     return x


def   A		(__A  :       Optional[int]  ,							__A  :       List[Any]			)  -> Union[str, Any]:







     """simple docstring"""


     UpperCAmelCase_		=				np.shape(a_			)
     if rows != columns:
          return np.array(()  ,							dtype=a_			)

     # augmented matrix
     UpperCAmelCase_		=				np.concatenate((coefficients, vector)  ,							axis=1			)
     UpperCAmelCase_		=				augmented_mat.astype('''float64'''			)

     # scale the matrix leaving it triangular
     for row in range(rows - 1			):
          UpperCAmelCase_		=				augmented_mat[row, row]
          for col in range(row + 1  ,							a_			):
               UpperCAmelCase_		=				augmented_mat[col, row] / pivot
               augmented_mat[col, :] -= factor * augmented_mat[row, :]

     UpperCAmelCase_		=				retroactive_resolution(
         augmented_mat[:, 0:columns]  ,							augmented_mat[:, columns : columns + 1]			)

     return x


if __name__ == "__main__":
    import doctest

    doctest.testmod()








"""simple docstring"""

def 						UpperCAmelCase   (						a_     ):





		'''simple docstring'''


		if not isinstance(a_,		a_     ):
				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 						UpperCAmelCase   (						a_     ):





		'''simple docstring'''


		if not isinstance(a_,		a_     ):
				raise TypeError('dynamic_lucas_number accepts only integer arguments.'     )
		lowerCamelCase      ,       lowerCamelCase	:     Optional[int]            =       2, 1
		for _ in range(a_     ):
				lowerCamelCase      ,       lowerCamelCase	:     Optional[int]            =       b, a + b
		return a


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

					testmod()
					_A				=					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)))






