





"""simple docstring"""


def 					lowercase__    (					lowercase_			)					->		int:




      """simple docstring"""

      if not isinstance(lowercase_		,lowercase_			):
            _UpperCamelCase      :	List[str]						=       F'''Input value of [number={number}] must be an integer'''
            raise TypeError(lowercase_			)

      if number < 1:
            _UpperCamelCase      :	List[Any]						=       F'''Input value of [number={number}] must be > 0'''
            raise ValueError(lowercase_			)

      _UpperCamelCase      :	Tuple						=       1

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

      return current_number


if __name__ == "__main__":
  import doctest

  doctest.testmod()







"""simple docstring"""


from collections import defaultdict
from math import ceil, sqrt




def 					lowercase__    (					lowercase_ = 1_000_000		,lowercase_ = 10			)					->		int:




      """simple docstring"""

      _UpperCamelCase      :	defaultdict						=       defaultdict(lowercase_			)

      for outer_width in range(3		,(t_limit // 4) + 2			):
            if outer_width * outer_width > t_limit:
                  _UpperCamelCase      :	Any						=       max(
                      ceil(sqrt(outer_width * outer_width - t_limit			)			)		,1			)
            else:
                  _UpperCamelCase      :	str						=       1

            hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

            for hole_width in range(lowercase_		,outer_width - 1		,2			):
                  count[outer_width * outer_width - hole_width * hole_width] += 1

      return sum(1 for n in count.values() if 1 <= n <= 10			)


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