

def    A_  (							a					,   a  ):

      """simple docstring"""



      if not isinstance(a					,   a  ):
            raise ValueError('iterations must be defined as integers'  )
      if not isinstance(a					,   a  ) or not number >= 1:
            raise ValueError(
                'starting number must be\n                         and integer and be more than 0'  )
      if not iterations >= 1:
            raise ValueError('Iterations must be done more than 0 times to play FizzBuzz'  )

      SCREAMING_SNAKE_CASE_				:  Any           =     ''
      while number <= iterations:
            if number % 3 == 0:
                  out += "Fizz"
            if number % 5 == 0:
                  out += "Buzz"
            if 0 not in (number % 3, number % 5):
                  out += str(a  )

            # print(out)
            number += 1
            out += " "
      return out


if __name__ == "__main__":
    import doctest

    doctest.testmod()









def    A_  (							a  ):

      """simple docstring"""



      if upper_limit < 0:
            raise ValueError('Limit for the Catalan sequence must be ≥ 0'  )

      SCREAMING_SNAKE_CASE_				:  List[str]           =     [0] * (upper_limit + 1)

      # Base case: C(0) = C(1) = 1
      SCREAMING_SNAKE_CASE_				:  Any           =     1
      if upper_limit > 0:
            SCREAMING_SNAKE_CASE_				:  int           =     1

      # Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i
      for i in range(2					,   upper_limit + 1  ):
            for j in range(a  ):
                  catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]

      return catalan_list


if __name__ == "__main__":
    print('\n********* Catalan Numbers Using Dynamic Programming ************\n')
    print('\n*** Enter -1 at any time to quit ***')
    print('\nEnter the upper limit (≥ 0) for the Catalan number sequence: ', end='')
    try:
        while True:
            lowerCAmelCase     :   Union[str, Any]						     =  int(input().strip())
            if N < 0:
                print('\n********* Goodbye!! ************')
                break
            else:
                print(F'The Catalan numbers from 0 through {N} are:')
                print(catalan_numbers(N))
                print('Try another upper limit for the sequence: ', end='')
    except (NameError, ValueError):
        print('\n********* Invalid input, goodbye! ************\n')

    import doctest

    doctest.testmod()






