
"""simple docstring"""






from __future__ import annotations

from collections import namedtuple
from dataclasses import dataclass




@dataclass
class 			_SCREAMING_SNAKE_CASE					:
			UpperCAmelCase_  :int
			UpperCAmelCase_  :TreeNode | None				 =						None
			UpperCAmelCase_  :TreeNode | None				 =						None




__UpperCAmelCase       =  namedtuple('CoinsDistribResult', 'moves excess')






def 						_snake_case       (	lowercase__		:  TreeNode | None		)  ->				int:


			'''simple docstring'''





			if root is None:
						return 0

			# Validation
			def count_nodes(lowercase__		:  TreeNode | None		) -> int:
						if node is None:
									return 0

						return count_nodes(node.left		) + count_nodes(node.right		) + 1

			def count_coins(lowercase__		:  TreeNode | None		) -> int:
						if node is None:
									return 0

						return count_coins(node.left		) + count_coins(node.right		) + node.data

			if count_nodes(lowercase__		) != count_coins(lowercase__		):
						raise ValueError("""The nodes number should be same as the number of coins"""		)

			# Main calculation
			def get_distrib(lowercase__		:  TreeNode | None		) -> CoinsDistribResult:

						if node is None:
									return CoinsDistribResult(0      ,     1		)

						lowerCAmelCase_     ,       lowerCAmelCase_    :List[str]			     =  get_distrib(node.left		)
						lowerCAmelCase_     ,       lowerCAmelCase_    :Dict			     =  get_distrib(node.right		)

						lowerCAmelCase_    :Optional[Any]			     =  1 - left_distrib_excess
						lowerCAmelCase_    :Tuple			     =  1 - right_distrib_excess

						lowerCAmelCase_    :Optional[Any]			     =  (
						    left_distrib_moves
						    + right_distrib_moves
						    + abs(lowercase__		)
						    + abs(lowercase__		)
						)
						lowerCAmelCase_    :int			     =  node.data - coins_to_left - coins_to_right

						return CoinsDistribResult(lowercase__      ,     lowercase__		)

			return get_distrib(lowercase__		)[0]


if __name__ == "__main__":
				import doctest

				doctest.testmod()













from math import ceil, sqrt





def      lowerCAmelCase_ (       __a = 1000000		)     ->		int:







  """simple docstring"""





  lowerCamelCase__:      Optional[int]			  =0

  for outer_width in range(3	,    (limit // 4) + 2		):
    if outer_width**2 > limit:
      lowerCamelCase__:      Dict			  =max(ceil(sqrt(outer_width**2 - limit		)		)	,    1		)
    else:
      lowerCamelCase__:      str			  =1
    if (outer_width - hole_width_lower_bound) % 2:
      hole_width_lower_bound += 1

    answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

  return answer


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

