




'''simple docstring'''
def 							snake_case_    (  a__						:      int       ,a__						:      int     ):



    """simple docstring"""




    __lowercase     							=		1.0
    for i in range(1       ,r + 1     ):
        ret *= (n + 1 - i) / i
    return round(a__     )
def 							snake_case_    (  a__						:      int     ):



    """simple docstring"""




    return choose(8 + n       ,a__     ) + choose(9 + n       ,a__     ) - 10
def 							snake_case_    (  a__						:      int     ):



    """simple docstring"""




    return sum(non_bouncy_exact(a__     ) for i in range(1       ,n + 1     )     )
def 							snake_case_    (  a__						:      int = 1_00     ):



    """simple docstring"""




    return non_bouncy_upto(a__     )


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










'''simple docstring'''
import numpy as np
def 							snake_case_    (  a__						:      Tuple     ):



    """simple docstring"""




    __lowercase							,__lowercase     							=		a.shape
    __lowercase     							=		min(a__       ,a__     )
    __lowercase     							=		np.eye(a__     )
    __lowercase     							=		a.copy()

    for k in range(t - 1     ):
        # select a column of modified matrix A':
        __lowercase     							=		r[k:, [k]]
        # construct first basis vector
        __lowercase     							=		np.zeros_like(a__     )
        __lowercase     							=		1.0
        # determine scaling factor
        __lowercase     							=		np.linalg.norm(a__     )
        # construct vector v for Householder reflection
        __lowercase     							=		x + np.sign(x[0]     ) * alpha * ea
        v /= np.linalg.norm(a__     )

        # construct the Householder matrix
        __lowercase     							=		np.eye(m - k     ) - 2.0 * v @ v.T
        # pad with ones and zeros as necessary
        __lowercase     							=		np.block([[np.eye(a__     ), np.zeros((k, m - k)     )], [np.zeros((m - k, k)     ), q_k]]     )

        __lowercase     							=		q @ q_k.T
        __lowercase     							=		q_k @ r

    return q, r


if __name__ == "__main__":
  import doctest

  doctest.testmod()




