"""simple docstring"""




from math import asin, atan, cos, radians, sin, sqrt, tan

_a      :		Any           =   6_37_81_37.0
_a      :		List[str]           =   6_35_67_52.31_42_45
_a      :		Tuple           =   6_378_137





def 							SCREAMING_SNAKE_CASE     (			_lowerCamelCase						:       float			,_lowerCamelCase						:       float			,_lowerCamelCase						:       float			,_lowerCamelCase						:       float	)       ->      float:
  _lowerCAmelCase						:       Any  			=     (AXIS_A - AXIS_B) / AXIS_A
  _lowerCAmelCase						:       List[str]  			=     atan((1 - flattening) * tan(radians(_lowerCamelCase	)	)	)
  _lowerCAmelCase						:       Dict  			=     atan((1 - flattening) * tan(radians(_lowerCamelCase	)	)	)
  _lowerCAmelCase						:       Optional[Any]  			=     radians(_lowerCamelCase	)
  _lowerCAmelCase						:       int  			=     radians(_lowerCamelCase	)
  # Equation
  _lowerCAmelCase						:       Dict  			=     sin((phi_a - phi_a) / 2	)
  _lowerCAmelCase						:       Tuple  			=     sin((lambda_a - lambda_a) / 2	)
  # Square both values
  sin_sq_phi *= sin_sq_phi
  sin_sq_lambda *= sin_sq_lambda
  _lowerCAmelCase						:       int  			=     sqrt(sin_sq_phi + (cos(_lowerCamelCase	) * cos(_lowerCamelCase	) * sin_sq_lambda)	)
  return 2 * RADIUS * asin(_lowerCamelCase	)


if __name__ == "__main__":
     import doctest

     doctest.testmod()








"""simple docstring"""







def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					__SCREAMING_SNAKE_CASE						   =				[0] * (upper_limit + 1)

					# Base case: C(0) = C(1) = 1
					__SCREAMING_SNAKE_CASE						   =				1
					if upper_limit > 0:
										__SCREAMING_SNAKE_CASE						   =				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(lowerCAmelCase_   ):
															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:
						a__				:							List[str]						=     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()
