
def 							a		(				snake_case__:   Tuple = 1_000   ):


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


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










import unittest

import numpy as np







def lowerCamelCase__  (     _lowercase					,		_lowercase					,		_lowercase					,		_lowercase = None					,		):

     '''simple docstring'''


     UpperCAmelCase_     :				Dict  =     np.shape(_lowercase )
     UpperCAmelCase_     :				Optional[Any]  =     np.shape(_lowercase )
     UpperCAmelCase_     :				Tuple  =     np.shape(_lowercase )

     if shape_a[0] != shape_b[0]:
          UpperCAmelCase_     :				Tuple  =     (
              '''Expected the same number of rows for A and B. '''
              f'''Instead found A of size {shape_a} and B of size {shape_b}'''
          )
          raise ValueError(_lowercase )

     if shape_b[1] != shape_c[1]:
          UpperCAmelCase_     :				List[Any]  =     (
              '''Expected the same number of columns for B and C. '''
              f'''Instead found B of size {shape_b} and C of size {shape_c}'''
          )
          raise ValueError(_lowercase )

     UpperCAmelCase_     :				Dict  =     pseudo_inv
     if a_inv is None:
          try:
               UpperCAmelCase_     :				Any  =     np.linalg.inv(_lowercase )
          except np.linalg.LinAlgError:
               raise ValueError(
                   '''Input matrix A is not invertible. Cannot compute Schur complement.''' )

     return mat_c - mat_b.T @ a_inv @ mat_b

class     __a(	unittest.TestCase  ):






    """simple docstring"""





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				str  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Any  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				List[str]  =     np.array([[2, 1], [6, 3]]    )

         UpperCAmelCase_     :				Tuple  =     schur_complement(_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE    )

         UpperCAmelCase_     :				List[Any]  =     np.block([[a, b], [b.T, c]]    )

         UpperCAmelCase_     :				List[Any]  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )
         UpperCAmelCase_     :				List[Any]  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )
         UpperCAmelCase_     :				Any  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )

         self.assertAlmostEqual(_SCREAMING_SNAKE_CASE  ,det_a * det_s    )





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				str  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Optional[int]  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				Optional[int]  =     np.array([[2, 1], [6, 3]]    )

         with self.assertRaises(_SCREAMING_SNAKE_CASE    ):
              schur_complement(_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE    )





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				Optional[int]  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Optional[Any]  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				int  =     np.array([[2, 1, 3], [6, 3, 5]]    )

         with self.assertRaises(_SCREAMING_SNAKE_CASE    ):
              schur_complement(_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE    )


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     unittest.main()