
"""simple docstring"""




from __future__ import annotations

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

# initial triangle of Koch snowflake
a		:       Union[str, Any]     						=       numpy.array([0, 0])
a		:       Dict     						=       numpy.array([0.5, 0.866_0254])
a		:       Union[str, Any]     						=       numpy.array([1, 0])
a		:       int     						=       [VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]



def     __magic_name__		(					UpperCamelCase     :		list[numpy.ndarray]							,		UpperCamelCase     :		int     )     ->						list[numpy.ndarray]:
    a__		=			initial_vectors
    for _ in range(UpperCamelCase     ):
        a__		=			iteration_step(UpperCamelCase     )
    return vectors



def     __magic_name__		(					UpperCamelCase     :		list[numpy.ndarray]     )     ->						list[numpy.ndarray]:
    a__		=			[]
    for i, start_vector in enumerate(vectors[:-1]     ):
        a__		=			vectors[i + 1]
        new_vectors.append(UpperCamelCase     )
        a__		=			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     __magic_name__		(					UpperCamelCase     :		numpy.ndarray							,		UpperCamelCase     :		float     )     ->						numpy.ndarray:
    a__		=			numpy.radians(UpperCamelCase     )
    a__	,     a__		=			numpy.cos(UpperCamelCase     ), numpy.sin(UpperCamelCase     )
    a__		=			numpy.array(((c, -s), (s, c))     )
    return numpy.dot(UpperCamelCase							,		UpperCamelCase     )



def     __magic_name__		(					UpperCamelCase     :		list[numpy.ndarray]     )     ->						None:
    a__		=			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__		=			zip(*UpperCamelCase     )
    plt.plot(UpperCamelCase							,		UpperCamelCase     )
    plt.show()


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    a		:       List[str]     						=       iterate(INITIAL_VECTORS, 5)
    plot(processed_vectors)




"""simple docstring"""




def     __magic_name__		(					UpperCamelCase     :		int							,		UpperCamelCase     :		list[int]							,		UpperCamelCase     :		int     )     ->						int:

    def count_of_possible_combinations(UpperCamelCase     :		int     ) -> 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(UpperCamelCase     )



def     __magic_name__		(					UpperCamelCase     :		int							,		UpperCamelCase     :		list[int]							,		UpperCamelCase     :		int     )     ->						int:

    def count_of_possible_combinations_with_dp_array(
        UpperCamelCase     :		int							,		UpperCamelCase     :		list[int]     ) -> 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							,		UpperCamelCase     )
            for item in array     )
        a__		=			answer
        return answer

    a__		=			[-1] * (target + 1)
    return count_of_possible_combinations_with_dp_array(UpperCamelCase							,		UpperCamelCase     )



def     __magic_name__		(					UpperCamelCase     :		int							,		UpperCamelCase     :		list[int]							,		UpperCamelCase     :		int     )     ->						int:
    a__		=			[0] * (target + 1)
    a__		=			1

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

    return dp_array[target]


if __name__ == "__main__":
    import doctest

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


