

import math
import sys

def      snake_case__      (   SCREAMING_SNAKE_CASE_		:   int      ):



     '''simple docstring'''





     if number != int(SCREAMING_SNAKE_CASE_      ):
          raise ValueError('the value of input must be a natural number'      )
     if number < 0:
          raise ValueError('the value of input must not be a negative number'      )
     if number == 0:
          return 1
     lowercase__   :       Tuple				=	[-1] * (number + 1)
     lowercase__   :       Tuple				=	0
     for i in range(1     ,					number + 1      ):
          lowercase__   :       Tuple				=	sys.maxsize
          lowercase__   :       str				=	int(math.sqrt(SCREAMING_SNAKE_CASE_      )      )
          for j in range(1     ,					root + 1      ):
               lowercase__   :       List[Any]				=	1 + answers[i - (j**2)]
               lowercase__   :       str				=	min(SCREAMING_SNAKE_CASE_     ,					SCREAMING_SNAKE_CASE_      )
          lowercase__   :       List[str]				=	answer
     return answers[number]


if __name__ == "__main__":
       import doctest

       doctest.testmod()




import numpy as np

def      snake_case__      (   SCREAMING_SNAKE_CASE_		:   str     ,					SCREAMING_SNAKE_CASE_		:   Optional[int]     ,					SCREAMING_SNAKE_CASE_		:   Optional[int]     ,					SCREAMING_SNAKE_CASE_		:   Union[str, Any]     ,					SCREAMING_SNAKE_CASE_		:   List[Any]      ):



     '''simple docstring'''





     lowercase__   :       List[Any]				=	int(np.ceil((x_end - xa) / h      )      )
     lowercase__   :       Dict				=	np.zeros((n + 1,)      )
     lowercase__   :       Dict				=	ya
     lowercase__   :       Union[str, Any]				=	xa

     for k in range(SCREAMING_SNAKE_CASE_      ):
          lowercase__   :       Optional[int]				=	f(SCREAMING_SNAKE_CASE_     ,					y[k]      )
          lowercase__   :       Dict				=	f(x + 0.5 * h     ,					y[k] + 0.5 * h * ka      )
          lowercase__   :       Any				=	f(x + 0.5 * h     ,					y[k] + 0.5 * h * ka      )
          lowercase__   :       List[Any]				=	f(x + h     ,					y[k] + h * ka      )
          lowercase__   :       Tuple				=	y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
          x += h

     return y


if __name__ == "__main__":
       import doctest

       doctest.testmod()

