


def     _UpperCamelCase     (a__   :list[int]   ,			a__   :int       ):






							"""simple docstring"""





							UpperCamelCase__				     =			len(a__       )
							UpperCamelCase__				     =			[[False] * (required_sum + 1) for _ in range(arr_len + 1       )]

							# for each arr value, a sum of zero(0) can be formed by not taking any element
							# hence True/1
							for i in range(arr_len + 1       ):
														UpperCamelCase__				     =			True

							# sum is not zero and set is empty then false
							for i in range(1   ,			required_sum + 1       ):
														UpperCamelCase__				     =			False

							for i in range(1   ,			arr_len + 1       ):
														for j in range(1   ,			required_sum + 1       ):
																					if arr[i - 1] > j:
																												UpperCamelCase__				     =			subset[i - 1][j]
																					if arr[i - 1] <= j:
																												UpperCamelCase__				     =			subset[i - 1][j] or subset[i - 1][j - arr[i - 1]]

							return subset[arr_len][required_sum]


if __name__ == "__main__":
					import doctest

					doctest.testmod()











from math import ceil, sqrt





def     _UpperCamelCase     (a__   :int = 100_0000       ):






							"""simple docstring"""





							UpperCamelCase__				     =			0

							for outer_width in range(3   ,			(limit // 4) + 2       ):
														if outer_width**2 > limit:
																					UpperCamelCase__				     =			max(ceil(sqrt(outer_width**2 - limit       )       )   ,			1       )
														else:
																					UpperCamelCase__				     =			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() = }""")




