



def 		a__      (		_UpperCamelCase      :			int				):

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

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

   __lowerCamelCase    				=     1

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

   return current_number


if __name__ == "__main__":
  import doctest

  doctest.testmod()











from __future__ import annotations

from math import ceil, floor, sqrt





def 		a__      (		_UpperCamelCase      :			int = 2_00_00_00				):
   __lowerCamelCase    				=     [0]
   __lowerCamelCase    				=     42

   for idx in range(1			,ceil(sqrt(target * 2				) * 1.1				)				):
      triangle_numbers.append(triangle_numbers[-1] + idx				)

   # we want this to be as close as possible to target
   __lowerCamelCase    				=     0
   # the area corresponding to the grid that gives the product closest to target
   __lowerCamelCase    				=     0
   # an estimate of b, using the quadratic formula
   __lowerCamelCase    				=     42
   # the largest integer less than b_estimate
   __lowerCamelCase    				=     42
   # the largest integer less than b_estimate
   __lowerCamelCase    				=     42
   # the triangle number corresponding to b_floor
   __lowerCamelCase    				=     42
   # the triangle number corresponding to b_ceil
   __lowerCamelCase    				=     42

   for idx_a, triangle_a in enumerate(triangle_numbers[1:]			,1				):
      __lowerCamelCase    				=     (-1 + sqrt(1 + 8 * target / triangle_a				)) / 2
      __lowerCamelCase    				=     floor(_UpperCamelCase				)
      __lowerCamelCase    				=     ceil(_UpperCamelCase				)
      __lowerCamelCase    				=     triangle_numbers[b_floor]
      __lowerCamelCase    				=     triangle_numbers[b_ceil]

      if abs(target - triangle_b_first_guess * triangle_a				) < abs(
          target - best_product				):
         __lowerCamelCase    				=     triangle_b_first_guess * triangle_a
         __lowerCamelCase    				=     idx_a * b_floor

      if abs(target - triangle_b_second_guess * triangle_a				) < abs(
          target - best_product				):
         __lowerCamelCase    				=     triangle_b_second_guess * triangle_a
         __lowerCamelCase    				=     idx_a * b_ceil

   return area


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






