




def 			_a   (		SCREAMING_SNAKE_CASE_							:       int  ):

					if not isinstance(SCREAMING_SNAKE_CASE_   ,   SCREAMING_SNAKE_CASE_  ):
										__lowerCAmelCase            = F"""Input value of [number={number}] must be an integer"""
										raise TypeError(SCREAMING_SNAKE_CASE_  )

					if number < 1:
										__lowerCAmelCase            = F"""Input value of [number={number}] must be > 0"""
										raise ValueError(SCREAMING_SNAKE_CASE_  )

					__lowerCAmelCase            = 1

					for i in range(1   ,   SCREAMING_SNAKE_CASE_  ):
										current_number *= 4 * i - 2
										current_number //= i + 1

					return current_number


if __name__ == "__main__":
					import doctest

					doctest.testmod()










from math import ceil, sqrt







def 			_a   (		SCREAMING_SNAKE_CASE_							:       int = 1_00_00_00  ):
					__lowerCAmelCase            = 0

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




