

from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
a__												=						numpy.array([0, 0])
a__												=						numpy.array([0.5, 0.8660254])
a__												=						numpy.array([1, 0])
a__												=						[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]



def   __UpperCAmelCase					(      __a :     list[numpy.ndarray]						,__a :     int      )     ->	list[numpy.ndarray]:

					"""simple docstring"""

					_a      :       Tuple   =						initial_vectors
					for _ in range(__a      ):
										_a      :       int   =						iteration_step(__a      )
					return vectors



def   __UpperCAmelCase					(      __a :     list[numpy.ndarray]      )     ->	list[numpy.ndarray]:

					"""simple docstring"""

					_a      :       Tuple   =						[]
					for i, start_vector in enumerate(vectors[:-1]      ):
										_a      :       str   =						vectors[i + 1]
										new_vectors.append(__a      )
										_a      :       Optional[int]   =						end_vector - start_vector
										new_vectors.append(start_vector + difference_vector / 3      )
										new_vectors.append(
										    start_vector + difference_vector / 3 + rotate(difference_vector / 3						,60      )      )
										new_vectors.append(start_vector + difference_vector * 2 / 3      )
					new_vectors.append(vectors[-1]      )
					return new_vectors



def   __UpperCAmelCase					(      __a :     numpy.ndarray						,__a :     float      )     ->	numpy.ndarray:

					"""simple docstring"""

					_a      :       Tuple   =						numpy.radians(__a      )
					_a ,   _a      :       List[Any]   =						numpy.cos(__a      ), numpy.sin(__a      )
					_a      :       Dict   =						numpy.array(((c, -s), (s, c))      )
					return numpy.dot(__a						,__a      )



def   __UpperCAmelCase					(      __a :     list[numpy.ndarray]      )     ->	None:

					"""simple docstring"""

					_a      :       str   =						plt.gca()
					axes.set_aspect('''equal'''      )

					# matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
					# y-coordinates as inputs, which are constructed from the vector-list using
					# zip()
					_a ,   _a      :       Optional[int]   =						zip(*__a      )
					plt.plot(__a						,__a      )
					plt.show()


if __name__ == "__main__":
			import doctest

			doctest.testmod()

			a__												=						iterate(INITIAL_VECTORS, 5)
			plot(processed_vectors)





'''simple docstring'''
def      UpperCamelCase_				(      A__  ,      A__  ,      A__						):

	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__						):

	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]
		a_      				=						sum(
		    count_of_possible_combinations_with_dp_array(target - item  ,      A__						)
		    for item in array						)
		a_      				=						answer
		return answer

	a_      				=						[-1] * (target + 1)
	return count_of_possible_combinations_with_dp_array(A__  ,      A__						)






def      UpperCamelCase_				(      A__  ,      A__  ,      A__						):
	a_      				=						[0] * (target + 1)
	a_      				=						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()
		lowercase__											=3
		lowercase__											=5
		lowercase__											=[1, 2, 5]
		print(combination_sum_iv(n, array, target))
