


from __future__ import annotations

from math import ceil, floor, sqrt




def     UpperCamelCase    (      _a = 2_0_0_0_0_0_0		)					->							int:
					'''simple docstring'''





					lowercase_       :list[int]     =       [0]
					lowercase_       :int

					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
					lowercase_       :int     =       0
					# the area corresponding to the grid that gives the product closest to target
					lowercase_       :int     =       0
					# an estimate of b, using the quadratic formula
					lowercase_       :float
					# the largest integer less than b_estimate
					lowercase_       :int
					# the largest integer less than b_estimate
					lowercase_       :int
					# the triangle number corresponding to b_floor
					lowercase_       :int
					# the triangle number corresponding to b_ceil
					lowercase_       :int

					for idx_a, triangle_a in enumerate(triangle_numbers[1:]       ,		1		):
										lowercase_       :str     =       (-1 + sqrt(1 + 8 * target / triangle_a		)) / 2
										lowercase_       :Tuple     =       floor(_a		)
										lowercase_       :Any     =       ceil(_a		)
										lowercase_       :Tuple     =       triangle_numbers[b_floor]
										lowercase_       :List[Any]     =       triangle_numbers[b_ceil]

										if abs(target - triangle_b_first_guess * triangle_a		) < abs(
										    target - best_product		):
															lowercase_       :str     =       triangle_b_first_guess * triangle_a
															lowercase_       :int     =       idx_a * b_floor

										if abs(target - triangle_b_second_guess * triangle_a		) < abs(
										    target - best_product		):
															lowercase_       :List[str]     =       triangle_b_second_guess * triangle_a
															lowercase_       :Union[str, Any]     =       idx_a * b_ceil

					return area


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





def     UpperCamelCase    (      _a       ,		_a       ,		_a		)					->							int:
     '''simple docstring'''





     def count_of_possible_combinations(_a		) -> int:
          if target < 0:
               return 0
          if target == 0:
               return 1
          return sum(count_of_possible_combinations(target - item		) for item in array		)

     return count_of_possible_combinations(_a		)




def     UpperCamelCase    (      _a       ,		_a       ,		_a		)					->							int:
     '''simple docstring'''





     def count_of_possible_combinations_with_dp_array(
         _a       ,		_a		) -> int:
          if target < 0:
               return 0
          if target == 0:
               return 1
          if dp_array[target] != -1:
               return dp_array[target]
          lowercase_       :Optional[int]     =       sum(
              count_of_possible_combinations_with_dp_array(target - item       ,		_a		)
              for item in array		)
          lowercase_       :List[Any]     =       answer
          return answer

     lowercase_       :Dict     =       [-1] * (target + 1)
     return count_of_possible_combinations_with_dp_array(_a       ,		_a		)




def     UpperCamelCase    (      _a       ,		_a       ,		_a		)					->							int:
     '''simple docstring'''





     lowercase_       :Optional[int]     =       [0] * (target + 1)
     lowercase_       :int     =       1

     for i in range(1       ,		target + 1		):
          for j in range(_a		):
               if i - array[j] >= 0:
                    dp_array[i] += dp_array[i - array[j]]

     return dp_array[target]


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   SCREAMING_SNAKE_CASE       :		Optional[Any]         =				3
   SCREAMING_SNAKE_CASE       :		Tuple         =				5
   SCREAMING_SNAKE_CASE       :		List[Any]         =				[1, 2, 5]
   print(combination_sum_iv(n, array, target))

